How can I add an outlet to a switch box?

It's likely that you have this scenario. You'd just connect the hot and neutral to the new outlet, and pigtail to the various switches. Blacks go out from the switches to the light or fan, and whites return to the neutral bundle.

Make Sure You Have GFCI Protection

There is already a GFCI in this bathroom

That is actually almost irrelevant. The rule isn't "you must have at least one GFCI-protected receptacle in the bathroom". It is two separate rules:

• You must have at least one receptacle in the bathroom (and powered only with other stuff in that bathroom or receptacles in another bathroom, but that is a capacity issue, which is a separate discussion, though it is a possible issue here because of "connected to my basement lights").
• Every ordinary receptacle in the bathroom (there may be exceptions if you have a ceiling receptacle for lighting, which you really shouldn't do, but I'm not sure...for practical purposes "all receptacles") must be GFCI-protected.

The protection can be provided at the receptacle (which I think is what you meant in your original statement), at another receptacle properly wired earlier in the circuit, with a GFCI-only device somewhere earlier in the circuit (in the same bathroom or another bathroom), or with a GFCI-breaker in the panel.

The usual setup is GFCI-receptacle because that is easy to retrofit and convenient. You have one of those. But this new receptacle must also be protected. Since this circuit is also powering lights, and since pulling power from the lighting circuit is far easier than connecting to the existing bathroom receptacle circuit, and since putting the entire circuit (i.e., including lighting) on GFCI has other problems (basement goes dark when GFCI trips), the simple solution is to install a GFCI-receptacle here instead of a plain receptacle. This is not just a good idea, it is code. In fact, someone used to "multiple GFCI-receptacles in a bathroom chained together and trip/reset together" would be left with a potentially fatal false sense of security seeing one receptacle with GFCI and one without.