When DS was born, he spent 6 weeks in the hospital. I was on maternity leave for 12 weeks, but I worried about the same thing. Figured we'd play it by ear. Fortunately, he didn't have a hospitalization until this past summer -- almost 3 weeks.
We both had adequate sick, comp and vacation time banked up. DH used comp time, I used a combination of comp and vacation. It got to the point near the end of DS' stay that we were looking at the possibility of 2 more weeks on IV abx in the hospital, so we were then going to have to take turns. DH was in the process of purchasing a new laptop to work from the hospital, I was going to head back for a few days and then tag off with him.
With dr. appointments locally, we take turns. For the ones in the city, we choose a Monday or Friday and make a long weekend out of it, though we HAVE made the trip in one day. We tend to use personal days, vacation and comp for those.
We both earn 1 1/2 days sick leave each month and are able to carry 120 days total. At the end of the year anything over 120 days, we get paid a lump sum for half the amount -- so we get a nice chunk of change and incentive to save up sick leave and use other paid time off instead.
Also, ever since a coworker got the flu and a secondary infection and was in a coma for several months several years ago, we also are allowed to donate sick leave or vacation to people who may need it. So that's another option we have if we run out.