By RAW, it does appear that you can continue to concentrate on a spell, even if the target succeeds on a saving throw.
Many Spells specify that a target can make a saving throw to avoid some or all of a spell’s Effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure.
The rules state that a spell specifies what happens if a creature succeeds or fails the saving throw. Using Bane as the example, there is no mention of what happens to a creature who succeeds the saving throw.
Up to three creatures of your choice that you can see within range must make Charisma Saving Throws. Whenever a target that fails this saving throw makes an Attack roll or a saving throw before the spell ends, the target must roll a d4 and subtract the number rolled from the Attack roll or saving throw.
Looking at the rules for concentration, the spell specifies how long you can concentrate on it for as well as specifying the reasons why you would be forced to drop concentration.
Some Spells require you to maintain Concentration in order to keep their magic active. If you lose Concentration, such a spell ends.
If a spell must be maintained with Concentration, that fact appears in its Duration entry, and the spell specifies how long you can concentrate on it. You can end Concentration at any time (no action required).
Normal activity, such as moving and attacking, doesn’t interfere with Concentration. The following factors can break concentration:
Casting another spell that requires Concentration. You lose Concentration on a spell if you cast another spell that requires Concentration. You can’t concentrate on two Spells at once.
Taking damage. Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your Concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
Being Incapacitated or killed. You lose Concentration on a spell if you are Incapacitated or if you die.
A creature succeeding on a save is not specified as a reason for a spells duration to end in the overall rules for spells. But there are some spells that do specify that a spell ends on a successful saving throw, such as Tasha's Hideous Laughter.
Tasha's Hideous Laughter
A creature of your choice that you can see within range perceives everything as hilariously funny and falls into fits of laugher if this spell affects it. The target must succeed on a Wisdom saving throw of fall prone, becoming Incapacitated and unable to stand up for the Duration. A creature with an Intelligence score of 4 or less isn't affected.
At the end of each of its turns, and each time it takes damage, the target can make another Wisdom saving throw. The target has advantage on the saving throw if it's triggered by damage. On a success, the spell ends.
If a creature succeeds on the saving throw for a spell that requires concentration, it appears that you can still concentrate on the spell as long as you are able, unless the spell specifies that it ends on a successful save.