You need to figure out what you are the most passionate about before making that choice, and this is often something you learn while getting a bachelors in engineering. Since you already have a B.S. in physics that may be hard.
A B.S. in physics CAN get you an engineering job as a EE or ME (mechanical engineering), but it is not easy since most places want to see the EE (electrical engineering), CpE (computer engineering), or ECE (electrical and computer engineering) degree title for EEs, and similar for MEs.
The best thing for you to do right now is to first pick a major based on what kinds of subjects you like.
There are some groups of engineering majors that are similar
1: electrical, computer, computer science
2: mechanical, civil, manufacturing, aerospace
3: biological, chemical
4: ?
Once you narrow down to a major start talking to people who work in that field, there are always diverse sub fields of each, electrical I can say from experience is incredibly diverse and I'm sure the same is true for the others. So once you find a subject your passionate about and enjoy talking about then you know what to do for a masters.
You also need to look at where you want to sit in the design phase when you get a job, or you may want to go into research (typically at a university).
Don't worry about the money, if you find a subject you really like then the skill will come with practice and then finding a job will not be so hard and most masters in engineering make plenty to support a family even in the US where living costs are higher than elsewhere. (that last sentence is not meant to start a social-political debate)