I made a similar transition a few years ago. I graduated with a BS in molecular/cell biology, did a considerable amount of (biology) research, worked for a while, took some time off, then decided I wanted to do mechanical engineering, so after taking a couple prerequisite math classes, applied for thesis-based MS programs in mechanical engineering.
The chemistry and math minors are good in that they prove you have some basic foundational quantitative ability, as is the fact that you’ve taken differential equations and linear algebra. What kind of research have you been involved in? Was it biological, or was there an engineering bent? Any research experience is good in that it teaches you about the research process, but unless it’s relevant to potential engineering research areas and topics, it won’t be terribly helpful in itself.
How did you do on the GRE? What was your GPA? While these are not the only factors that will be considered for graduate-level admission, they are strong indicators of success in graduate-level coursework (though not necessarily indicators of success in graduate-level research). These numbers will be especially important for you, since you’re coming from a non-quantitative background.
Do you have experience with programming?
On a more general note, to put it bluntly, a biology major with your background will have a relatively difficult time being truly competitive for most ME Master’s programs, but it can be done. I didn’t have any real programming knowledge or engineering coursework or engineering-related experience, but was able to get into some excellent thesis-based MS programs on account of high GPA/GRE, an understanding of the research process, and the ability to express my motivations and make a case for myself very well in my personal statement. And, probably, also a little bit of luck. However, if I had to redo it, I would do a lot of things differently. For instance, I didn’t take any engineering courses before starting my MS program. Sure, I’d studied a number of topics–mechanics, thermodynamics, fluids, etc.–on my own to some extent, but this doesn’t compare to the depth and breadth of engineering courses that engineering majors take. I was sorely underprepared relative to my classmates, which led to a stressful first year. And though I did ultimately survive the deep end of the pool (and learned a ton of stuff very quickly in the process, by necessity), I would not recommend this approach. Taking a number of basic engineering courses will not only make your application significantly more competitive/realistic, it will also ease the transition.
Another thing you should definitely do: learn to program if you don’t already.
Why do you want to get an MS in ME, if you don’t mind my asking?