How do you deal with hot weather on a boat when there is no air conditioning?
Get an air conditioner?

Seriously, though, I recognize that that is not a cruising solution or even one for anchoring out. You did not give the context, so I took the shot.
This year, we made it well into June, almost July, before putting the ac on the boat. We tried an air scoop for the fore hatch (which works incredibly well when the wind is blowing - well worth the $30 or so, even cheaper if you make one) and a few well placed, well aimed fans.
Those days with 111 degree heat indexes take their toll, though. The killer is no-wind days. It is little help to have fans, as you are just moving near 100 degree air in the boat, which is not all that pleasant.
It actually averages about 10 degrees or so cooler right on the coast (we are about 20 miles inland). So, 95-98 here is mid to upper 80's where you might more likely want to anchor out without ac this time of year. Also, down there, wind is almost always blowing at least a little bit, and again, that wind scoop has to be experienced to be believed.
So, it could be done without too much discomfort. We don't push the ac on board here until we are CONSISTENTLY in the upper 90's with no real cool-off at night AND once we hit the summer 'doldrums.'
We do have a few liveaboards here in the marina that have no ac on board. Some of them spend a fair bit of time in the air conditioned Captain's Lounge. Too much time, maybe.
All of this gets back full circle to something that I repeat a LOT => GET OFF THE BOAT. A few weeks ago, before we shipped our ac on the forehatch, we would leave during the hot part of the day. Library, aquarium, mid-day movie, or just out doing 'our thing.' By spending time in the local community, one is not couped up in the boat all day anyway.
For me/us, the critical metric is getting cool enough (and non-humid enough) at night to get a good night sleep.
We anchored out one night a few years ago that was a record-setter and the hottest night of the year - nearly 110 degrees actual thermometer temp (heat index even higher) and we had a t-storm, so all hatches closed. It was a long, hot, soggy night. That can be a 'sticky' situation fer sure.
