Given what a bustling tourist mecca it is these days, it's hard to believe that the city of Cancun, with its long stretch of inviting beaches, didn't even exist until 1974, when it was founded as a planned community specifically to draw tourists. And that mission has been quite successful, as the tourists are now often quite thick along the many fine beaches. But if you do a little scouting you can still, amazingly, find many nooks and crannies where you can enjoy the beach without the crowds, and sometimes even in near isolation. Playa Tortugas is often uncrowded, yet has many activities available, and is a good launching point for an exploration that may lead you to even more private spots nearby. Meanwhile, if the crowds don't bother you, then of course there are also plenty of places on the beach where you can find a festive atmosphere.
If you go:
If you are staying somewhere outside the expensive Hotel Zone, it's very easy to get an inexpensive bus or taxi ride to the beaches. Admission to the beach is free.
Links:

21.148561991844204, -86.790893684829
Cancun, Quintana Roo 77500 Mexico
Mailing address:
Mexico
Phone Numbers:
Cancun Convention and Visitors Bureau
+52 (998) 881-2745
+52 (998) 881-2745