Yeah, many state parks and public beaches have "No Alcoholic Beverages
Allowed" signs all over the place. I put Coca-Coca and Dr. Pepper in
the cooler when I go to those places.
After all, it would be embarrassing to be arrested by a lifeguard.

It wouldn't be a lifeguard arresting you. If they are like Minnesota, the
state parks and larger park systems have their own police officers and the
others can bring in the local authorities quickly.

However, they have better things to do than bust people for alcohol unless
they are causing a problem.

While on a camping trip at a state park, we had a group site (secluded, away
from the main sites). We went for a late night hike and talked with a state
park ranger along the way. He casually asked if we had any alcohol, then
said that if we did, we shouldn't carry it around (we wouldn't anyway) or be
overly loud.

I volunteer with a local (Three Rivers Park District) park district. The
"park patrol" is around for information and first aid. After appropriate
training, we use the trails (skate, bike, hike, ski) and watch for people
who need assistance. Usually it is just to help people figure out where
they are or where something is.
However, if we see a problem, we will call the dispatcher for the park
police.