Just because you have mice doesn't make you a slob. It is YUK! YUK! YUK! But, most of the time it is out of your control, especially if you live by forest preserves or open fields.
Yes a cat makes a difference, however, you say you are having another baby so if you do get a cat, Dad will have to be the one to clean the kitty litter because this could be a medical threat to you. You may also want to consider that if you do have a cat and you are in a community with close neighbors, your neighbors may not appreciate your cat in wandering in their yard all the time. I was THAT neighbor who had my neighbor's cat in my yard every day, in the winter. It never occurred to me, until Spring, that my "large flower bed" was his kitty box. And yes... that was gross.
You may also want to get a few of those "sonic mouse traps" that plug into an outlet and emit an offensive noise only heard by these critters. I use these in my garage and it helps to deter them. Coupled with the easier (Tomcat) mouse traps, that look like a larger, pinching clothespin, this should help. These are my preference. I hate setting the old fashioned mousetraps.
Even if you get a cat, you will probably want to determine the possible source of entry and reduce their access.
If your basement is unfinished, have someone go down in your basement during the day, with the lights off so that you can see any tiny cracks, spaces in joists, window wells, etc... so that these can be filled with steel wool or foam (canned) insulation. If there is old, fiberglass insulation, you may want to consider removing/replacing it so that you can see where these cracks are.
If this doesn't work (or if you don't get a cat)... have the place exterminated. It's amazing the small spaces that these varmints can squeeze through.
Good luck.