There are many opportunities for internships in government for you to gain experience in public policy or international affairs.
I've interned in non-profits and government and found both experiences very valuable. I majored in Political Science as an undergraduate and my master's degree is in International Commerce & Policy. I spent a semester in college in Washington D.C. and was able to intern subsequently in a federal agency.
Consider relocating to D.C. or a state capitol for internships with representatives or senators or with a legislative committee. If you have this opportunity in D.C., intern with a committee or representative. Interning with a committee will get you more substantial legislative work, especially if you know you want to focus in a certain area such as homeland security or healthcare. The offices of representatives are much smaller staffed than senators so it's likely you will have more opportunities to get real experience. Federal agencies are another great way to get government experience. Keep in mind, some of them have long application periods, such as the State Department, so you will need to start applying well before you want to intern.
Look for internships with a local or state government agency. Also, volunteer to do some campaign work; knowing how the political machine operates is valuable. Volunteering in this capacity is a great way to network too, as you'll meet a lot of people involved in government or who will be involved in government in the future.
Consider interning for a non-profit too. I see your primary goal is interning for the government, but I just want to make you aware that many non-profits also do public policy and international affairs work.
Good luck to you and I hope you find this information helpful! Please feel free to reach out if you have questions.