If they said drilling the floor will solve the problem, it's not a plumbing issue, it would be ground water, probably seeping in through the foundation. Drilling holes in the concrete floor allows the water to soak into the ground underneath, but this is actually horrible for the stability of the floor. The water can compact the soil and cause low spots. The right way to do it is to cut a sump hole at the lowest point in the floor with a pump in it to pump the water into your drain. The pumps have floats that turn them on when the water comes up. You can install a backup pump that runs in a car battery for power failures if they're common in that neighborhood. All together it should only cost you about $1000 to have someone cut the floor and dig the hole and set it all up.
Chances are your real estate agent didn't inform you of the lead laws (they rarely do). In MA, if the property isn't deleaded when you buy it, you are responsible for the past tenants that may have been lead poisoned while living there unless you delead within 90 days of closing.
The state is trying to get every rental property in MA lead safe, and they figured this was the best way to do it.
You also cannot rent to someone with a child under 6 unless you have a lead certificate on file with the state. If you rent to a couple with no kids and they have a baby, you are then required to delead the unit with the added expense of putting the tenants up in a hotel during the process.
If a unit is a 2 or 3 bedroom, chances are you'll end up having a family apply. You cannot deny the family because of the lead paint either, that's considered discrimination (you're refusing them because they have a kid) and is against the anti-discrimination laws of the state. If there are kids under 6, you NEED to have the unit deleaded and have a lead certificate on file.
If the unit has been inspected, but not deleaded, you also cannot do any construction or remodeling in the unit, they consider that unauthorized deleading and it will prevent you from ever getting a lead cert. The best you would be able to get is a Letter of Environmental Protection, which tells everyone you did illegal work then hired a licensed deleader in to fix it. It generally increases your insurance rate and decreases your tenant pool (who wants to rent from a shady landlord that does illegal work?).
You can use this in your negotiations on any future properties you look at. If it doesn't have a lead certificate on file, it IS going to cost you to delead, the state has pretty much made it mandatory at this point (at least if you want to avoid the liability of a lead paint lawsuit from a past tenant). Don't let that scare you though, I've had clients negotiate $20k off the selling price, and I only charged them $10k for the deleading, so it can actually work in your favor for it not to have been done yet.
On the plus side, when you delead the property it will actually increase your tenant pool and the rent you'll be able to charge them. A deleaded unit is more valuable to a tenant then a non-deleaded unit, even if they don't have kids under 6, they still like to know they are safe from lead paint issues. Anyone can get high blood lead levels, the law only specifies under 6 because it affects the developing brain more severely then a it affects a fully developed brain.
If you want to message me any addresses you want to look at, I can check the database and see if there was ever an inspection done on the property for you. I'll be able to tell if there's a lead cert on file already too. Maybe you'll get lucky and it's already been done.
Good Luck on your investment!