I'm a business analyst with a similar background (intelligence analysis, liberal arts, visual presentations and written reports). At my technology company we have found that SQL/DBA work can be added as you go and that "seeing the big picture" is the more essential skill.
I'd look for:
1) A tech company
2) One that is growing (see Inc 500|5000 list for too many leads to even chase)
3) Basic fluency – take some courses on codeacademy, codeschool, etc
It really depends on the type of Business Analyst you want to be - what kind of work you want to do. There are courses and certifications for Business Analysts you can look in to which might be helpful. You may want to give serious consideration to a tool like Crystal Reports, which can connect to any database and generates its own SQL. After all, the point isn't to be a Database Admin, it's to get at data and analyze it.
If you have the option of taking one course in Database Management or Analysis, do it. The basic principles will take you far.
If you really want to learn databases and SQL I would recommend starting with MS SQL Server Express. It's free, relatively easy to use and is well documented.