How can I tell if the stitches in my finger are getting infected?

Redness and swelling. Generally speaking, a wound is infected if it looks "angry". Redness that is persistent and worsening. Swelling that is worsening and very tender to touch. If the infection remains untreated the wound may open or loosen and pus may be expressed from the wound.
Redness or drainage. If there is significant redness, drainage or the pain seems to be getting worse, see your surgeon to determine if the sutures are infected.