Does the Reverse Gravity spell affect creatures moving into the area after it is cast?

The reversed gravity affects all creatures in the area

The spell clearly says:

This spell reverses gravity in a 50-foot-radius, 100-foot high cylinder centered on a point within range.

It does so for the entire duration. The creatures do not fall upward due to some other, unrelated magical effect forcing them to. They fall upward because gravity is reversed.

If you happen to be in the area when it is cast, you always get to make a saving throw to grab onto a fixed object in reach to avoid the effect. (If there is no such object, and you are not anchored, you're out of luck).

If you enter later, it's up to the DM if they ask for a saving throw. In case you weren't aware of the effect, this could be for realizing it and stepping back, like with a pit trap. In that case you wouldn't need a fixed object.

Yes - but in the "standard" way

The duration of the spell is 1 minute. What does the spell do for one minute?

This spell reverses gravity in a 50-foot-radius, 100-foot high cylinder centered on a point within range.

The casting time of the spell is 1 Action. What does the spell do when cast?

All creatures and objects that aren't somehow anchored to the ground in the area fall upward and reach the top of the area when you cast this spell.

This distinction is because casting the spell represents a specific exception to typical conditions - it is forced movement - and this part of the spell explains that exception, specific over general. Once the spell is cast, movement is not forced: gravity works as gravity normally works in the game, only in the reverse direction. No special instructions are needed because this is the general game state and the DM knows how to apply background gravity. If a creature enters the area of reversed gravity after the spell is cast, then the DM simply applies the new gravity direction to their motion, but without any special rules added by the spell.

If a creature would normally fall when not supported by the ground, they would fall to the new 'down' when they entered the area. The DM could choose to allow a Dex save, but is not required to - this should be treated as any other 'sudden fall' situation, such as if PCs were on a platform that collapsed.

But if a creature wouldn't normally fall when not supported by the ground, entering the area of the spell would not make them fall. No Dexterity save would be needed, because they wouldn't fall regardless.

One way to see this difference is with reference to a flying creature. If a flying creature entered the area of reversed gravity after the spell was cast, it would sense the change in gravity, adjust, and continue its flight motion uninterrupted - having a fly speed permits the creature to not fall when it is not on the ground. But if the spell was cast on an area with a flying creature included in it, the creature would fall upward until it reached the top of the area, with forced movement as directed by the spell, specific over general. Nothing about having a fly speed renders a creature immune to effects that force it to fall, as this spell does when it is cast. After the spell was cast and the flying creature had fallen, it could resume its normal movement in the area of the reversed gravity.