First reaction: Noooo! Indeed, traveling by taxi in Brussels, with the one exception that I mention below, is simply a no-no. Because it is just too expensive. I only have one taxi experience so far, which was to return from the airport at 1 a.m. to my house (near Schuman), with no traffic at all and a trip that took like 10 minutes and it costed me 37€. Ouch! And I have talked to other people with taxi experiences in Brussels and all of them were sure to declare it was an one time only thing, given the price they ended up paying.
As for a fact, public transportation in Brussels works really fine and you have a lot to choose from: metro, buses and trams. During the day I believe it is possible to reach almost any place you want with these 3 options. However, during the night they don't work, so you sometimes may feel tempted to take a taxi. On one hand, you have some night buses, and I think it's preferable to walk all Avenue Louise in the middle of night with 0º to take a Noctis bus in Port de Namur (been there, done that!) in order to avoid paying the price of two nice dinners to a taxi driver.
However, there's one taxi service that won't make you have to eat pasta for a week to compensate that late return from a party with friends. It is called Collecto and it is a service of collective taxis that can be used all days from 23h to 7h in Brussels. This service has more than 200 places of departure, usually located where there are STIB stops. You take the taxi in one of these stops and it will take you to any address in Brussels for a fixed price of just 6€ (in case you have a STIB pass, it is only 5€) per person. Here you can read on how to reserve these taxis. The only disadvantage is that you need to set up a time for them to pick you up and when you're in a party or a dinner you can't exactly calculate when you're going to leave... but you can request the service 20min before wanting to use it, so even if you request it when you feel like leaving, you'll only have to wait 20min until the taxi arrives, which is not much considering you will still have to walk until the stop. I haven't used this service so far, as I have been able to get around with the buses only, but if I ever need a taxi again at night I will definitely go for it!