What does "would" mean in this context?
Man: People are starting to stare. Can you keep your voice down, please?
Woman: Absolutely, we wouldn't want them to think we're not happy.

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Hi there, 
I assume that the context for this dialogue is that the woman is complaining about something that other people present ('them') have done or failed to do?
If so, then;
'wouldn't want' indicates that the speaker is referring to the possible future consequences (of speaking with a loud voice)
'don't want' indicates that the speaker is referring to the definite present consequences (of speaking with a loud voice)

Note also that this structure is used for irony. So the woman may be contradicting the man, not agreeing with him.

Hope that helps, Helen

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it softens it.

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@necip You're right, I figured it out from the context. But, why didn't she use "don't" instead of "wouldn't"?

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İt means like "dont" in this context

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