According to information I have received May is a little dryer, however the last time we were there mid May some animals are just coming out from the winter. We did NOT see very many bears and the Salmon are not running yet. We did see moose, mountain goats and a lot of sea life along with plenty of birds. The waterfalls are great in May due to snow still melting.
My choice would be September. The last time we were in Alaska, the weather was beautiful and didn't see a mosquito. We even got to see Mt McKinley for 5 full days!!!
It's always a gamble on either side, since they are both considered to be off-season. May rates tend to be better. At least according to what's available right now. I believe Princess is close to being sold out for this September - at least when we are going on 8 September. You'll see a lot more salmon in September.
Definately mid May, after the begining of Sept. the weather muliplies several times over for likely bad weather- temps 40-50's expect rain several days, dusk at 7pm. I have gone both times you mention and only go late Sept for the very cheap clearance sales - have been less than $500 the last 2 years, but I also make an earlier trip for my favorite time to go- mid to late August. May is dryer, long days and warmer temps 45- 60 possible. It still requires layers and rain gear where ever you go. I do not recommend Sept overall after 9 trips.
The poster is refering to mid Sept and this is not a recommended time to go. Also there is far more salmon mid July to beg of Sept- not later. As I said I do not recommend Sept travel for the increased possibility of poor weather as the month goes on. Also a real negative with dusk at 7pm mid Sept. Rain is far more likely as well with cooler temps. This is from first hand experienc, going twice a year.