Honey bees are 0.3 to 0.8 inches in size, and bumble bees are 0.4 to 0.9 inches, so a one inch mesh should not hinder them too much. (Especially if there are pollen/nectar producing flowers inside...) Unless you are growing some exotic plants that require large moths for pollination, you should be fine.

Our biggest problem is ground squirrels, so we have to line under the planting bed area and up the sides about 3 feet with one inch mesh chicken wire to keep them out. And even that only lasts a few years before we have to replace it.

I have squirrel, rabbit and bird problems. So, we just built a 25' X 50' garden enclosure.
We used the 1" chicken wire and it works great. FYI, we found that they have stucco wire here at Lowes that is basically galvanized 1" chicken wire for about $50 for a roll that measures 150' X 3'. It ends up being a lot cheaper.
Another thing is that we were able to get blemish pipe for a great deal. We used, 2 3/8" for the uprights and 1 5/8" for the top of the structure. All of the connecting hardware was readily available from the same fence company where we got the pipe.
As cuestararoble said, the 1" is plenty big for bees to pass through. I even had a 3' long gopher snake go through mine. He had no problem at all, he slid right through. I may have to go up the sides with some hardware cloth with smaller openings :)