1. Have talked with your profs 1-on-1 so they can actually get a sense of your personality and strengths (this should be past tense before you ask them, but if it's a lower key internship it's less of a deal if they don't know you very well. But you can't ask them if they don't have any sense of who you are). Ask them if they can write you a supportive letter. Anyone can write a letter. I've written one that, between-the-lines, reads, "this student is a kind soul, but is a flake and isn't ready for the high pressure demands of your internship," because he was desperate, and forgot the stuff and was last minute, keeping with his personality he'd shown me throughout the semester as a D student who couldn't turn something in or show up on time to save his soul. Your prof's professional integrity is on the line. They're going to be tactful, but they're not going to lie on your behalf. The whole point of a recommendation from a trusted member of the community is that they're supposed to be writing with integrity. So make sure they can write you a good one.
2. bring the context that they're writing to them. The more blind they are to the end goal the harder the letter will be to write. This is bad for your internship and for your prof's attitude when they see you've put them into a frustrating position where they have to make a blind move, so to speak. See if you can print off the 'about' pages on their website, and a job description of the internship for each one.
3. Don't just ask the prof to write the letter, tell them why you want to apply. Again, give them resources to help for a successful letter. Have a larger life trajectory that they can see this fitting into.
4. Ask ASAP because they're busy people. It's customary to provide self-addressed, stamped envelopes so all they have to do is drop it in the mail.
5. Bring it in a folder with everything printed off instead of electronic so they can lay it out on their desks and type.
6. If they do it, a token measure of gratitude is nice. It doesn't have to cost money it just has to be a gesture of thanks.