Easy... Contact companies you would want to work for and offer to intern or work for free for a few days/week under someone with that experience. Soak in the info. Help them in other areas. Ask a lot of questions as if you are completely new to the field.

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.

There is much more to being a business analyst SQL or databases. So, if I were you, why would I want to become a business analyst? One of the main reasons I want to be a business analyst is because I always measure the success of my work through the positive impact I leave on the company's goals, product's improvements, and client's needs. Another thing I love about being a business analyst, is the chance this role gives me to work with people from different departments in the company who have a variety of great expertise, and always have interesting perspectives to add to the table. I think that one of the remarkably interesting aspects of a business analyst's job is understanding the core need of the client and working with the development team to fulfil that need. The beauty of this role is that business analysts are always evaluating their product's performance, which gives the company a great advantage in terms of staying on top of the latest trends and implementing clients' feedback with every new step. In general, I believe that this role has so much to offer in terms of a long term professional career, and allows one to continuously learn new techniques to tackle complex business problems and find innovative solutions in today's technology driven economy.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath