You're in luck, because the power levels of the old Basic D&D games are similar to those of 5e.* I'm going to illustrate my answer with examples from B5 Horror on the Hill, but you can apply the principles to any of your Basic D&D Modules.
1. Replace what monsters and NPCs you can from the 5e Monster Manual
This will save you a lot of work, and will also give you a feel for the differences between the two editions. The 5e Monster Manual is quite comprehensive, although you may have to be a bit creative - so there are no Neanderthals in the MM, but there are "Tribal Warriors".
2. Convert the rest of the monsters and NPCs
As I said, the power levels are fairly similar, so there's not too much work to be done here. The main tasks are:
- Use positive armor class: Starting with 3e D&D, armor class goes up rather than down from 10, so AC 3 in Basic D&D = AC 17 in 5e.
- Decide some Strength, Dexterity and Constitution scores and apply the modifiers to combat rolls, hit points and saving throws. Don't sweat this too much - it's really just a case of deciding on a number from -4 to +4 to give the monster strengths or weaknesses.
- Give the monster or NPC skills or special abilities which make sense for its role in the adventure, so if it is meant to surprise the PCs, give it Dexterity (Stealth) +4 or +6
- Convert saving throws using the PHB. So if a monster's stat block in B5 says it has save F4 (Fighter level 4), give it proficiency (+2) in Strength and Constitution saving throws, as per the 5e PHB Fighter entry (PHB 71).
3. Make sure you understand how Ability Scores and Saving Throws work in 5e, including setting DC, and passive Perception
You can read up on these in the 5e core books you have. Most of what you need to start is in Chapter 7 of the PHB, and the DMG gives more detailed advice if you need it.
So for example this trap from B5:
Unless the party is actively looking for traps, the first two characters stepping onto the trap doors trigger their opening. Characters immediately behind the first pair must successfully check their dexterity or fall into the pit (they must roll their dexterity score or less on a d20).
I would interpret this in terms of a DC 15 Wisdom (Perception) check to spot - so unlikely to be spotted with passive Perception, and then a DC 10 Dexterity saving throw so as not to fall in - about even chances, with high Dexterity making a difference. Note that in 5e both of these rolls mean trying to roll the target number or above on a d20.
4. You can get more detailed with the DMG
An alternative way of dealing with traps in particular would be to take the most similar ones from the examples in the Dungeon Master's Guide, so 'Hidden Pit' (DMG 122) seems the best fit here.
The same goes for magical items - either leave them as they are, or take the nearest alternative in the DMG.
There are other curiosities about B5, for example the (to modern eyes excessive) use of percentages to determine if things will happen, but these can actually make things quite fun for the DM in terms of the unexpected happening. But apart from the above changes, most of which can be made on the fly, you are good to go!
*You can test this by comparing the "Wandering Monsters on the Surface of the Hill" table (B5, 5) to the "Wilderness Encounters" table of the 5e Starter Set (The Lost Mine of Phandelver, 27) - the entries are comparable and in some cases almost identical.