When is it appropriate to use 'the' instead of a possessive determiner?

Both are grammatically correct. "The" is much less common and might have a negative connotation based on "the wife." However, the somewhat condescending "the wife" or "the husband" seems to be used in place of "my wife" or "my husband," which is different from the example you heard, where "the" is replacing "her."

It can be said both ways.

Let me give you an example:

That is her computer.

But you can also say it this way:

That is the computer.

Therefore:

Wow, the husband must be so proud.

That sentence is just as correct as this:

Wow, her husband must be so proud.

Both are valid, but would be used in slightly different contexts. The normal usage of determiners governs the choice here.

You use 'the' to signify that the determiner phrase refers to a referent already in scope. In this case, if the husband was mentioned previously (perhaps as part of a family), using the is fine:

I was talking to a couple earlier, and the wife told me she was having a baby. The husband must be so proud.

In this case "the husband" and "the wife" are in scope, since they constitute the "couple* mentioned.

If the referent is not in scope, you should use a different determiner to specify which husband you're talking about. So

I was talking to Alice earlier, and she told me she was having a baby. Her husband must be so proud.

Since "Alice" has been mentioned, the possessive determiner her suffices.

Exceptions

There are some cases where you can refer to family members with the without their being explicitly mentioned. For example, "the kids" is normally understood to refer to the speaker's own children. As others mentioned, "the wife" (or "the husband) can refer to one's spouse; this usage is condescending to the spouse.