Borås is best from late May to early September, when the parks are in leaf, the river paths are usable, and the city's outdoor café terraces are open. Summer days this far north stretch on past 10pm; the city stays awake with them.
The Borås International Sculpture Biennial runs every two summers and is a good excuse to time a visit — the temporary installations are scattered across the city centre alongside the permanent collection. The No Limit street-art festival typically takes place in early September and is when new murals go up; if you want to watch them being painted, that is the week to be in town.
Winters are dark, damp, and cold rather than dramatically snowy — Borås is too close to the Atlantic to get much of the deep frozen weather of inland Sweden. The museums and indoor sights make a visit perfectly viable in any season; just expect short days and grey light from November through February.