So my boyfriend is leaving to basic right as I'm leaving on an exchange to Germany for 5 months. What's the best way to keep contact?

Considering that your boyfriend is going to be in basic, letters are really your only real option.

As others have stated, use any means available. Social media, skype, facebook, video calls with messenger, email, and even snail mail. All these options and more. One thing I would stress is that you both understand that it may not be convenient to talk at times, you may be busy with studies or he may be busy with basic/AIT. Or you each may be getting to know the area or people around you. There is nothing wrong with that, especially if you have a solid foundation of trust and respect in your relationship. As much as you may mean to each other, give yourselves and each other space enough that you each may continue to grow in your respective areas. I am currently on assignment away from my wife and family for the past few months now but I make it a point to call or video her once a day. There are times we talk for 10 minutes because she's busy with the kids, church, or her friends or I may be exhausted from work or need to spend time with my soldiers. There are days we talk for 2 or 3 hours, and I enjoy that time with her as it lets us connect on a deeper level. Best advice I can say is go day by day, savor your time together, and don't be too disappointed those times you can't connect.

Write letters the old fashioned way. Give each other the space to focus on your respective goals and responsibilities. If the relationship is strong and built on trust, you shouldn’t have any problem six months from now when you see each other again.