I’m currently a sophomore, but when I’m a junior I want to apply for the senate page program. I would definitely apply for the summer session of 2016-2017. How difficult is it to get in?
summer is definitely harder to get in. You usually see more people with connections doing the summer program. Yes there are more people in the summer, but you get a heck of a lot better of an experience doing one of the semesters. Being there for three weeks hardly lets you get accustom to the capitol hill environment. Then again, this is coming from a Fall semester former Page! 
It seems like the competitiveness of the program depends on where you live and who you apply through more than anything else. It never hurts to apply though!
I agree with @il0510 . I was a summer page and it was a lot harder since people with connections ran the place. Out of the 32 residential pages who attended, I would say around 7 of us got in without connections so apply early! Doing the semester program would be ideal, but it depends on your situation and if your senators can sponsor. I applied for both semester and summer programs, but ended up doing the summer since Senator Rubio and Senator Nelson don’t sponsor pages during the school year.