Safety

I don't know what your husband claims, and being neither female nor 19 I am perhaps not the best authority.
This refugee thing has 0% affect on 99.9% of the people, it is mostly newspaper hysteria, which gets more hysterical the further away you get. European cities are a lot safer than those in the US.
The city I live in is 2-3 times larger than Salzburg (which is a lovely place). Last night (Sunday) I was out and about in the city centre after dark. There were, as usual lots of people about. Including females in their early twenties or younger, alone or in groups of 2 or 3, just giggly and enjoying themselves without a care in the world, as usual (oh, to be young!). Friday and Saturday night are busier
Nobody drives (you can't in the city centre), they would all be going home by tram or bus after a few drinks and maybe a bit of clubbing.
This is normal in a medium sized European city. Your daughter will probably have a lot of fun.

There have been many discussions of late on this topic, they can be searched at the top of the page using a couple of keywords. See link below to one such thread. In another thread, here is part of a comment that I made:

Practicing smart travel by not being an easy target for pick-pocketing, and incorporating awareness of goings-on around you, where you are and at what hour, etc, can pretty much ensure you'll move through Europe unscathed. It is important to not over state the safety question, imo.

In your daughter's case, the best advice I can give you is to encourage her to practice situational awareness and be cautious with alcohol in public settings. I suspect that the "dangers to women" are not as serious as some may think.

https://community.ricksteves.com/travel-forum/germany/travel-concerns

My 11 year old goes to/from school everyday day in Vienna using regional train and subway. I'm totally comfortable with this as Austria is about the safest place in the world. I bet it's safer than where your daughter goes to school. I'd be a lot more concerned about the frat parties at her current school.

I would add that I feel totally safe as a woman here - and I work with refugees.

First of all I make no judgments about some one else's threshold of comfort or classification of threat levels. Its just way too subjective.

My daughter is currently studying in Bonn, Germany. When she left my advice to her was to avoid certain situations and groups of people so much as it was possible to do so. Otherwise just be aware of your environments and surroundings; and act accordingly. Given that and my perception of what was going in on in Germany I felt pretty good about the situation. A few weeks ago I went to visit her. Frankly, I didn't like what I saw and I am now worried. So this isn't second hand supposition. My concern is based on what I saw and experienced. I don't want to play the "its safer there than here" game; because quite frankly no one know where I live or where most of those posting here live. So its a nonsensical statement. I would much rather my daughter walk at night from her class to her home at her US university than I would in her current situation. Hands down, no competition. I saw exactly the city block where my daughter went to school, the route she had to take to the train station (she had to ride the train to her home) and the activity in the train station. If I had to do it again, I might have just said no. And the insinuation that all US college students get involved in some sort of illicit Frat activities while the same or equivalent does not exist in Europe is a pretty naïve view on life.

What I saw in Bonn I also saw in Cologne. On the other hand, my daughter will be spending a week in Budapest in March. I expect her to be out late at night. I expect her to be walking home after dark. I have no concerns. So its an issue of location. I don't know about Salzburg or Vienna.

It's one of those things you have to figure out on your own. Don't listen to me or anyone else as we all have individual biases; even those that live there. To expect otherwise would be like assuming since I am from Texas and I support B. Sanders, then all Texans must support him. To one person Austria is tranquil and safe (and I suspect for the most part it is), to thousands others something else is going through their minds as gun sales and gun carry permit applications have more than doubled in recent months. Now the percentage of Austrians owning guns is very nearly the same as the percentage of Texans owning guns. So who is right? I thought I knew with Bonn and I think now I was wrong. My gut about Salzburg is that it is probably as safe as any upper middle class American community. But you need to find that comfort level for yourself. Google News might be a good place to start.

I have spent quite a bit of time in Salzburg over the years and I really can't imagine a safer place for her to spend an academic year. It's a small city, with a huge tourist presence and a very active, lively, local population. It has a small core with outlying suburbs like most cities. You can drive across the whole place in 10 minutes, including the suburbs. I haven't noticed any run down areas where I would be wary of her being.
As for the refugee situation, I really can't say but I haven't heard of any incidents in Austria that would make me hesitate to send my child there, especially Salzburg.

It would help to have more information on what your husband "claims". No one can give you an absolute guarantee that your daughter won't have any problems, but of course that's true at home as well. I was in Salzburg a few months ago and didn't notice any problems with the refugees at that time. I talked with a few of them (as much as the language barrier would allow) and all they wanted was to get to Germany, so they're not staying long in Salzburg.