For getting around the San Jose urban area, there are various different public transport options available, including plenty of buses as well as a localised commuter rail service. The historic centre of San Jose is also reasonably walkable, and you should have no problems finding taxis.
Yet for heading out of San Jose and into the Costa Rican jungle, then some sort car is going to be really useful, although you might prefer to opt for an organised or private tour instead.
Do you need a car in San Jose? Yes, it can be useful.