May, June or September

The Wikipedia page for Dubrovnik has a summary weather chart. It shows that, in terms of average high and low temperatures, June and September are very similar. May is cooler. Early May and late September would be expected to have a good bit more rain than June. In the interior city of Zagreb, September is more on par with May temperature-wise but is the driest of the three months, and it's cooler at night than the coast during all those months.

You might want to take a look at some day-by-day historical weather data on wunderground.com to see the range of what you might expect. Averages only take you so far.

My wife and I went in mid-May. The weather was generally warm throughout our trip, but we had a couple days with clouds and rain where the temperature dropped a bit. Our day in Mostar, Bosnia was sweltering (and I'm told that it only gets hotter later in summer). The evenings were usually cool. It will only get warmer in June, but I don't know by how much.

Regarding crowds, you will have cruise ship crowds to deal with no matter when you visit. May is less crowded than June, but it's a matter of degrees. You should still take crowd avoidance measures (show up to Plitvice early before the tour busses arrive, plan Old Town Dubrovnik for morning and late afternoon, etc). In May, it was easy to get tickets for the catamarans and long distance busses. The catamarans were only about half full. Obviously, as you get more into the season, the fuller the local transportation becomes.

Not sure how May/June compare to September, but I'm sure someone here will have an opinion. :)

Late May or early September.

Where in Croatia do you want to go? Istria, being much further north has a much shorter season than Dalmatia.

Early May, you may find in smaller resorts that not everything is open fully. Local schools start their summer holidays c 15 June, so places will get busier after this date, but nowhere near as busy as late July and August.

If you want to swim in the sea, you need to go June onwards and September will be the warmest water.

The weather often breaks at the end of the third week in September in Dalmatia, but Istria had good weather into October this year.

Dubrovnik gets many cruise ships, so if visiting here, try to avoid busy sailing days.

The weather there can be very localised - I have been on Hvar in the sunshine looking over to Split on the mainland which was experiencing lightning and rain.

I would travel either the first two weeks of June or the first two weeks in September.

June and September is fine for the beaches. Warm spots are: Hvar, Korcula and Dubrovnik. To skip crowds you need to stay Islands and pick exotic beaches.