We can be much more helpful when we see what you have in mind. There are various reasons we might use one or another form. Also, it makes a difference whether you want the word to function as a pronoun or as an adjective.
Like onemorning85, I would use which, and for the same reason.
I speak American English. I would like to know what a speaker of British English would say. Sometimes I hear them use "who" to refer to the government, because they are thinking of the government as a group of people.* However, I don't know how they decide whether to do this, so I don't know whether they would do it in this example.
* In case anyone is interested, here are some earlier threads on the issue:
Well, as a moderator is asking for opinions and not just closing the thread . . . The problem with looking at previous threads is that they will go all round the general issue, but it might be hard to find a specific answer.
In this particular sentence I would definitely say 'which'. It is only the collective government that has the good reputation, not the individuals that compose it. Singular agreement is the only one possible in:
The government was formed after the May 2010 election.
The government was dissolved in preparation for new elections.
The government consists of Liberal Democrats and Conservatives.
I would use plural (but could also use singular) when 'the government' could be replaced by 'the members of the government':
The government have agreed on a new policy.
The government are keen to resolve the issue without delay.
This question is slightly removed from the "government - singular or plural" question, though the two are connected.
In BE, we are happy to consider government as singular or plural - as explained above. etb has explained the rationale for our choice.
If we find ourselves needing a relative pronoun for government, it depends on the same rationale. Speaking of government as a set of individuals, we use who; speaking of government as an impersonal entity, we use which.
In AE, as I understand the opinions expressed here, government is singular and therefore requires which, not who.
I can't imagine either AE or BE tolerating: The government, who has a good reputation, has not solved the problem yet.
So the answer to the post #1 question, as amplified in post #3, is straightforward: the sentence considers government to be singular - use which, not who