Mürren works in two seasons and is often closed in between. The summer season runs from late June to mid-October, when the Allmendhubel funicular and most hiking trails are open, the meadows are in flower, and the cable cars run full schedules. This is the best time for walking and for clear views — though cloud can settle into the valley at any time, so check the webcam before making the trip.
The winter season runs from roughly December to March, with skiing on the Schilthorn slopes and in the connected Jungfrau ski area. Mürren is significantly quieter than the bigger Bernese Oberland resorts, which is either a virtue or a drawback depending on what you want. The shoulder periods (November, early April) see reduced services and many restaurants closed; check ahead before visiting.