No droop, but they will catch wind. I had an Outdoor Research Seattle Sombrero and gave it up as it was too hot and way too floopy-- really useless in the wind.
If you want a hat with a really stiff brim, the Filson "Tin" hats are great. They make one with a chin strap too: http://www.filson.com/product/index.jsp?productId=2156785
Duluth Trading Co has some knock-offs too. http://www.duluthtrading.com/search/searchresults/98754.aspx?feature=Product_2
The other hat I like is the military "boonie" hat. Any surplus store will have a bunch and they come in khaki, camo, green, and black. Should run about $11-$12. They come with a lanyard, made of rip-stop cotton. Fold up it and stuff in a pocket, biner it on your pack, throw it in the washing machine, wet it in a stream and cool off your noggin, use it for picking berries... I like khaki flavor-- better in the sun and not as scary as the other colors.