Human cultures remain linguistically divided. Some converse in Spanish, others in Mandarin, Arabic, Swahili, etc. Every alien race has their own equally broad panoply of languages and dialects.
Most individuals know only their mother tongue, and rely on machine translation. Modern portable computers allow anyone with a few hundred credits of equipment to enjoy seamless real-time translation of alien languages, courtesy of handheld PDAs, computers in clothing or jewelry, or sub-dermal implants. Without fast and accurate translation, galactic trade and culture would not exist.
Governments provide subsidized software, updated through the public extranet "on the fly", often as users approach spaceport customs facilities. Even the batarians, who isolated themselves from galactic society nearly two decades ago, take pains to provide up-to-date glossaries and linguistic rules, though most suspect that this is only so they can continue exporting propaganda.
It is still considered broad-minded and practical to be able to speak without machine aid. Children often take courses in alien language, and most races can speak the simplified artificial "trade tongue" with little difficulty.
Some species must rely on machine translation to interact with the rest of the galaxy. Hanar, for example, cannot reproduce the spoken language of any humanoid species, and other races cannot reproduce hanar bioluminescence without mechanical aid. Newly discovered or obscure races don't have machine translation available until the linguists have had time to study them.