Weekday visits are dramatically better than weekends. The town that is pleasantly atmospheric on a Tuesday afternoon becomes almost impassable on Saturday. If you are staying in Taipei and planning a day trip, Monday through Thursday is far preferable; public holidays are worse than regular weekends.
Weather is variable on the northeast coast, which catches weather systems coming off the Pacific. Autumn (September to November) and winter (December to February) bring persistent drizzle and cloud, which actually suits Jiufen's atmosphere rather well — the mist in the lanes and the lanterns glowing through low cloud are more evocative than clear skies. Spring can be excellent. Summer is hot and humid, and the peak tourist season brings maximum crowds.
Go in the late afternoon regardless of when you visit. The lanterns light up between 4 and 5pm, and the hour before dusk is the best time to be on Shuqi Road. The last tour buses leave around 9pm, after which the town becomes much quieter and the authentic atmosphere reasserts itself.