I live in Northern California and have had problems with spiders and spider bites in several homes. I have used Ortho's Home Defense and it works great. I use a "nuclear" method. I spray around the house on the outside and every room in the house over the course of 2 hours. The spiders disappear as do ants.
I would immediately spray every room in your apartment.
It would bbe great if you can spray the areas outside your apartment as the spiders may be infested at your neighbors as well. However, I realize that you have limited control in an apartment building, . Make an inquiry to your neighbors to see if they have the problem, you can join forces with them in confronting the apartment management. Next, I would take photographs of your baby and the spiders and get in touch with your apartment manager and show him the evidence of the spiders. Your neighbors should accompany you at this meeting. If he refuses to do anything about them, escalate it to your local health officials.