Trouble is, if the latex is peeling, then there's a good chance that the floor is 'transpiring', ie., moisture is making its way up through the floor and 'forcing' the loose. By simply adding a different paint, you will make the problem worst.
What you might want to do is if you have an area that's about 2' x 2' w/o any paint, do a test for moisture by COMPLETELY taping a sheet of plastic to the floor and waiting a couple of days. If there's a 'measureable' amount of moisture under the plastic, you may want to let the floor 'breath'. This means that you want the paint to ultimatly come off the floor. In fact, if you can help it, do so. If the basement is completely dry AND you have very little condensation under that plastic, you could consider sealing it. You want to talk to your local Retail Paint Expert and describe the problem once the existing paint is off the floor. I don't know how well a sealer will 'work' if there is any latex on the floor, even if it's just residue. That's why you want to talk to someone who knows about paint and concrete and concrete 'sealers'.
Others may have some ideas but that's the best I could offer. Obviously, your case is one for 'others' to ponder as to why one shouldn't paint/seal a floor that's breathing moisture (assuming this is what it's doing.) It's hard to tell exactly what's going on w/o seeing too.