Protecting your teleportation circle
Complete the circle with a higher-level slot
This is the obvious implication of the linked question: the spell level of the permanent teleportation circle is the spell level of the final casting that makes the circle permanent. Higher-level spells are more difficult to dispel.
Assuming a large organization is setting up the circle, the highest-ranking magic user in that organization may have access to spell slots higher than 5th level. If so, they should use the highest level spell slot they can to complete the circle. For flavor, this practice might be enshrined in a "christening ceremony" of sorts, where the leader of the organization (who also happens to be the highest-level caster) is called upon to complete the work of setting up the circle.
(Obviously, if your table uses a different rule, e.g. setting the level of the circle to the average spell level of all castings involved in the setup, then the strategy will be different. But the core idea of "make it the highest level possible" remains the same.)
Limit physical access to the circle
However, that doesn't stop a dedicated adversary, who could potentially make many attempts to dispel the circle, eventually succeeding by chance even if it was cast at 9th level. (Or a high-level adversary may be able to cast dispel magic at a high enough level to guarantee dispelling the circle.) So, any permanent teleportation circle will be guarded. The DMG makes this point as well, in the section on how teleportation circles fit into the world:
However, since every teleportation circle is a possible means of entry into a city, they’re guarded by military and magical protection.
The DMG gives a different reason for guarding the circle: to protect the city from attackers using the circle to bypass the city walls. However, it is also logical that the organization in control of the circle will want to protect their investment from being ruined.
To that end, it makes sense that they will not let just anyone see the circle. Dispel magic requires unobstructed line of effect to work, and preventing adversaries from ever having line of effect to the teleportation circle is the most effective countermeasure against tampering. The circle will be placed in somewhere that is not visible from publicly accessible areas of the building. Anyone trying to tamper with it will have to sneak past the guards and other defenses (including magical alarms and wards) in order to get line of effect. The circle may even be in its own dedicated room, protected by a locked and reinforced door (which will likely be further reinforced by an arcane lock spell).
Dispelling the circle is not the worst an enemy can do to it
Lastly, it's worth noting that if an enemy knows about the teleportation circle, dispelling it may not be the best course of action. As the DMG notes, teleportation circles are potential entrances. To me, it seems more likely that an enemy would attempt to gain knowledge of the circle's sigil sequence and attempt to leverage that knowledge, rather than just attempting to ruin the circle. Depending on the nature of the enemy, they could sell the sigil sequence to the highest bidder, hold the organization ransom, or use it themselves to help their evil organization infiltrate the city. But any possible value they could derive from knowing the sigil sequence is destroyed if the circle is dispelled.
Of course, limiting access to the circle as described above will also limit an adversary's ability to learn its sigil sequence.
Dispel magic is not the only way to destroy a circle
In a similar vein, an enemy that wishes to destroy a teleportation circle need not employ magic of any kind. As the spell says:
Each such circle includes a unique sigil sequence--a string of magical runes arranged in a particular pattern.
Physically disrupting the runes in the sigil sequence (using, for example, a pickaxe) will prevent it from functioning, effectively destroying it. Obviously, this isn't as easy or stealthy as casting dispel magic from 120 feet away, but this method is available to anyone who can swing a pickaxe. Any circle must be protected against physical attacks as well as magical ones.
Ok, but let's create a public teleportation circle anyway
However, in light of all of the above, we can try to figure out what might be required for a city to maintain a public teleportation circle. Here is one possibility:
First of all, the circle must not be placed within the city walls. Anyone could show up at this circle, and they must be checked by the guards before they can enter the city itself. The most likely placement for this circle would be at a guard house just outside the city walls. Since a permanent teleportation circle only serves as an arrival point, there is no need to have a public entrance to the circle, even if the circle is nominally "public". Most likely, the circle will be in a well-guarded room in the guard house itself. Hence, despite being "public", it will not be possible for a random spellcasting passerby to walk up and cast dispel magic on the circle.
Anyone arriving at the circle will be immediately ushered out of the room with the circle and into a holding room, where they can be subjected to a thorough inspection. All spellcasting is forbidden in the room containing the circle, and the circle will be guarded at all times not only by city guards, but also by spellcasters who can cast counterspell. If anyone attempts to cast any spell near the circle, they will counter it, and the guards will arrest the caster. (There is no need to make an exception for casting teleportation spells, since the circle is only an arrival point.)
If the circle needs to have "business hours" outside of which is is not accessible, this can be accomplished by casting private sanctum to cover it with a teleportation ward at close of business, and then dismissing the sanctum the following morning when opening up. (But consider that "business hours" might be problematic for people teleporting in from other time zones.)
The above description is not the only way to secure a public circle against magical tampering, but it makes the point that it is well within the capabilities of a city to maintain such a circle with minimal risk of having it dispelled by either random acts of magical vandalism or a dedicated antagonist. The guiding principle is that an unattended teleportation circle protected only by spells and other magic wards is not feasible, nor would any city want such a circle to exist. The circle must be guarded by people, not just spells.
In terms of worldbuilding, it's clear that a public teleportation circle like this requires quite a lot of upkeep, even more so than a private circle. At the very least, it would require full-time wages for both guards and at least 3 5th-level spellcasters (to cover the circle 24/7 with 8-hour shifts). Is it worth it for a city to pay this cost? Maybe it is, if providing teleportation access encourages increased long-distance trade in rare items (which the city can easily tax, since they control the teleportation circle).
The best modern analogy to a teleportation circle might be an international airport: people can arrive from anywhere in the world at any time bringing with them all manner of dangerous items, and the associated infrastructure is expensive and delicate, so the security practices must reflect that. (Let's ignore the questionable effectiveness of the security at real-life airports. Your fantasy security can be as effective or ineffective as you want it to be.)