This question, of course, is thinking primarily of 1 Timothy 2:12 (NIV): I do not permit a woman to teach or to assume authority over a man; she must be quiet. (Other verses especially in Titus 1 could also be cited.)
I think the simplest answer to this question is to go to Romans 16. There we find Phoebe (a deacon and a patron of Paul), Prisca (a co-worker of Paul's), Mary and Tryphena and Tryphosa and Persis (who all worked so hard in the service of those in Rome that Paul knows of it, even though he has never been to Rome), Junia (outstanding among the apostles), Rufus' mother (who was also a mother to Paul), and Julia and Nereus' sister (who with Philologus, Nereus, and Olympas seem to have led a group of the Lord's people).
Remembering there was no formal church structure in the first century (some structure, certainly, but not nearly as organized as in later years), those who believe in ordaining women recognize that Paul appreciated these women and their ministries alongside his own. This then leads us to believe that the Timothy passage cannot be taken as a general prohibition but may best be seen within the context of that passage where the gentleness of both men (1 Tim 2:8) and women (1 Tim 2:9 and 15, where modesty and self-control are the same word in Greek) are being emphasized. For more on this, see “B. L. Merkle’s Interpretive Method Applied to 1 Timothy 2:8-15.” Paper Presented at the 2010 Annual Conference of The Evangelical Theological Society, Atlanta, GA. November 18, 2010.
For further reading, see this roundup of resources.