Anytime is a good time to visit Kauai, however there are a couple of considerations including weather and airfare.
Generally, it rains more in the winter months than the summer months. In the Poipu area, weather is usually not a factor as it stays mostly sunny and dry year around. On the North Shore, the winter rains and high surf can be more of a factor, especially for those hiking the Kalalau/Hanakapiai Trail. The trail becomes very muddy and slippery during rains, making the hike to Hanakapiai Beach and beyond more difficult. Also, heavy rains can cut off access to Hanakapiai Beach and trap hikers on the far side, requiring evacuation by the Coast Guard. For the North Shore, the best times to visit Kauai are probably from May to September. (But keep in mind it can rain, and rain a lot, at any time of the year, especially on the North Shore.)
As for airfare, the summer months are the most expensive. Unfortunately, over the last several years the airlines have reduced capacity to most all travel destinations, driving up prices. Airfare to Kauai is now several hundred dollars higher in many cases for roundtrip tickets. There are two ways to get the cheapest airfare. Book many months out before demand causes remaining tickets to rise in price, or wait until one to three months before you wish to visit and check to see if airlines have lowered prices because of reduced demand (which sometimes happens during the spring and fall months). The only problem with the latter strategy is finding lodging at shorter notice.
Overall, the best month to visit Kauai is August (in our opinion), which is usually drier island-wide. For both price and weather considerations, May is a great choice.
(FYI: Whale watching season is from November through May, with the peak months being February and March.)
A quick analysis bears this out. We reviewed roundtrip airfare from LAX to Lihue for the months April to December 2015. We selected the first full week (Sunday to Sunday) in each month. (Cheaper results may be obtained by traveling during the week.) Here are the results:
From LAX to Lihue Airport (Roundtrip)
April 5 – April 12 Nonstop $528
May 3 – May 10 Nonstop $524
June 7 – June 14 Nonstop $630
July 5 – July 12 Nonstop $736
August 2 – August 9 Nonstop $736
September 6 – September 13 Nonstop $628
October 4 – October 11 Nonstop $628
November 1 – November 8 Nonstop $628
December 6 – December 12 Nonstop $632
(Prices may vary)