Do open fractures in finger usually get an infection, even when they're bandaged?

Finger Fracture: Open finger fractures have a higher rate of infection then closed finger fractures but a lower rate of infection than open fractures in other areas of the body. They are usually treated with irrigation, debridement, fixation as needed and oral or IV antibiotics.

Although an open fracture in finger has more risks for infection but still occur only occasionally if a proper treatment is given immediately. You need to see a hand surgeon or visit an emergency room asap.