Be not unequally yoked with unbelievers: for what fellowship have righteousness and iniquity? or what communion hath light with darkness?
Let's read it in context:
14 Be not unequally yoked with unbelievers: for what fellowship have righteousness and iniquity? or what communion hath light with darkness? 15 And what concord hath Christ with Belial? or what portion hath a believer with an unbeliever? 16 And what agreement hath a temple of God with idols? for we are a temple of the living God; even as God said, I will dwell in them, and walk in them; and I will be their God, and they shall be my people. 17 Wherefore Come ye out from among them, and be ye separate, saith the Lord, And touch no unclean thing; And I will receive you,
Let's look at some Bible commentary.
John Gill's Exposition of the Bible has some comments:
This seems to be an allusion to the law in (Deuteronomy 22:10) and to be a mystical explanation of it; and is to be understood not as forbidding civil society and converse with unbelievers; for this is impracticable, then must believers needs go out of the world; this the many natural and civil relations subsisting among men make absolutely necessary; and in many cases is both lawful and laudable, especially when there is any opportunity or likelihood of doing them any service in a spiritual way:
So it does not mean not to associate in any way with unbelievers, and it would especially not forbid doing good works with them. Why not fight abortion hand-in-hand with prochoicers who want to see abortion reduced or restricted?
rather, if any particular thing is referred to, it is to joining with unbelievers in acts of idolatry; since one of the apostle's arguments to dissuade from being unequally yoked with unbelievers is, "what agreement hath the temple of God with idols?" and from the foregoing epistle it looks as if some in this church had joined with them in such practices; see (1 Corinthians 10:14,20-22) . But I rather think that these words are a dissuasive in general, from having any fellowship with unbelievers in anything sinful and criminal, whether in worship or in conversation;
Seems like it's pretty self-evident: Don't join unbelievers in evil-doing. And I've not seen any "common ground" efforts that involve prolifers in any way promoting, facilitating, or even condoning abortion. So, looking at Gill's commentary, there's no reason not to join with unbelievers in good works -- nor even to join in good works with people who have vast areas of sin in their lives. We're just not to join them in sin.
We can also look at People's New Testament:
14-18. Be not unequally yoked with unbelievers. The figure is drawn from the prohibition of Deut. 22:10 and Lev. 19:19. The meaning is that Christians are not to pair off with unbelievers. All intimate associations are forbidden. The primary reference is to intermarriage and to association in heathen festivals. All close fellowship with unbelievers is included. Saints should choose Christian alliances and associations.
This indicates that the idea of being "yoked with" the unbelievers is to enter into close partnership with them, such as marriage or in their festivals. But there's nothing here to forbid doing good works with an unbeliever.
The issue seems to be the reason for associating with the unbelievers. After all, you have to get close enough to them to preach the Gospel in the first place!
I think many people confuse "common ground" with compromise. And I've never espoused compromise.