The best time to hike the Routeburn Track is during the Great Walk season - which runs from October to April. This assumes you are hiking end-to-end. Some day walks on the Routeburn are possible in the Spring and Autumn shoulder months. As a general rule, the summer months of December to February, and also March, are the prime hiking months as there are mild-warm conditions and normally less rainfall. It is worth remembering though, the Routeburn track traverses mountainous valleys in the Southern Alps, so the landscape is prone to rainfall at any time of year. Hikers need to pack for all seasons.
When weighing up the best month to do the Routeburn Track, you may prefer the shoulder months of November and April if you prefer less of a crowd. Another factor to weigh up is availability of accommodation in Queenstown, where you'll likely stay pre and post walk. November and early December as well as February and April typically offers the best availability and options. The period from mid-December to mid-January often books out early, and is subject to peak rates and minimum stay periods. Be sure to book your Queenstown accommodation at the same time as your Routeburn Hike to secure the best options.
If you're planning a day hike on the Routeburn Track, there are some sections that are able to be walked most of the year, such as the Key Summit hike. The Routeburn Falls end of the track is also accessible in the off-season but heavy rain and snow can cause degradation to the track and small land slips if conditions are bad, so always check the track conditions and weather before planning your trip, or ask your advisor for tips.