If you play free gigs, you'll certainly get your name out there. But you'll become known as someone who doesn't need payment for their services.
I'm not talking about playing a fundraiser for a cause or donating your time to a charity event. In these cases, you're the initiator choosing to donate your time and skill. I'm referring instead to being asked to play gigs like weddings or pit orchestras where your time should be compensated just like the other support crew that makes the event happen.
If you aren't sure if you should be playing a gig for free, take a look around. Is the caterer getting paid? Is the cleaning crew being told that while there's no budget to pay them, this will really help them get their names out there? The point of going to music school, or any other higher education program for that matter, is to gain specialized skills. You can do things that not everyone can do, and when you're hired to do those things, you should be paid because it's a job.
Some organizations have a small budget, and this makes it hard for them to pay everyone what they deserve. But even a token stipend, coupled with the admission that more would be given if possible, is better than asking someone to work for free.
Something I really feel strongly about is the concept of unpaid internships. Unless you're going to receive valuable training that you can only get by working there, don't do it. It's just not reasonable to expect someone to put in a full-time workweek and receive no pay; the bottom line is that it's financially impossible unless you're independently wealthy.
When I was working toward my master's, I briefly interned in the box office at a Boston-based musical organization. The internship was unpaid, but I figured it would be worth it because I would get to learn ticketing software and how a nonprofit works, and maybe make some contacts in the musical community.
I ended up only staying for two weeks. They brought me on shortly before a big concert, and things were simply too busy at that point to train a new person on how to use the ticketing software. Instead, I was put to task alphabetizing old files.
Alphabetizing. For free. With an hour-and-a-half commute. While going to school full time.
They did have me help out the day of the concert by working the will-call desk, but this only consisted of sifting through a small box of tickets that were set aside for people to pick up the day of the concert and checking patrons' IDs to make sure they were who they said they were when they claimed their tickets.
Once I realized I already knew the alphabet and its various usages, I let them know I wouldn't be continuing the internship.
I recognize that not everyone's experience is the same; some people work unpaid internships or play free gigs that turn into great careers. But I think the concept of asking for something in return for nothing is something that needs to change.