Very important. If she is going to not take the new job, she needs to refuse the job offer without working on it a minute! This will greatly increase her chances of being approved for unemployment insurance, as they would look at it as she voluntarily quit the job for a good job related reason, that they eliminated her position and offered her something that paid less. This is generally going to be looked at as a good job related reason to quit.
It should, of course, be a significantly less good salary. And you should realize, as cbg pointed out, that unemployment insurance at its very best probably does not pay as much as working, even at a reduced pay rate. You have to quickly, before the changed position is to begin, make a decision.
Is it better to work at something that pays less and be looking for another job that pays more while working, or to quit all together, have a fairly good chance of being approved for unemployment benefits, and with this reduced (or possibly not there at all, there's no way to guarantee she'll be approved) amount of money coming in and lots of free time, she'd be looking for another job too.
Unemployment insurance in the best of cases is also temporary, regular unemployment is only good for six months or less, and there are at this time some extensions funded by the government, but these could go away at any time with no guarantees.
But if she does accept the new position, works at it even for a short time (ten minutes, even!) and then quits, there is virtually no chance she will be approved for unemployment insurance, because she has accepted the job, with its reduced pay and conditions, and then has voluntarily quit it by her own choice.
In all my years I have never understood people who quit a job with no prospect of something new to go to because "I couldn't live on what I was making." I always want to ask them what they'll be living on now.