Could taking prednisone for my cough affect my getting over the flu by depressing my immune system?

Unlikely: For Prednisone to suppress the white blood cell function of the immune system it takes high doses for at least 7-10 days. A short course of Prednisone will not affect the immune systems ability to function normally. On the other hand, high doses or long term daily or every other day use could affect immune function.

Yes and no: This is always a balancing act. Yes, high doses of Prednisone can decrease the responsiveness of your immune system, but the sleep disruption from a severe cough can be just as immunosuppressive. Particularly if you have asthma, it is helpful to take steroids to treat the cough; once you're able to get a full night's rest for a few days, you'll be able to fight off the virus despite the prednisone.