First of all, whether you're seeing it or watching it, "basketball game" requires an indefinite article; it should be either "a basketball game" or "the basketball game."
AmE seems to have its own rules when it comes to discussing sports. If the people in your quote are planning to attend a basketball game in person, they would say "We're going to the basketball game." If they plan to view the game on TV, they would say "We're going to watch the basketball game." (Actually, in normal conversation, they'd be very specific, saying "We're going to watch the Pistons game" or something along those lines.)
Using "see the basketball game" is correct grammar but not standard usage.