1) Contribute to (or just read the code of) open source projects in order to improve your coding skills
2) Read up on existing research in topics you're interested in
I understand the desire to be efficient and kill two birds with one stone. However, finding a good, active, community built research project might be a bit difficult. Even if you find it, you may find that opportunities for you to contribute to it are very few.
Contributing to open source projects will give you a good opportunity to understand the open source contributing process (which you'll need to know when you do find that open source project you want to contribute to). It'll also allow you improve you coding skills while you're at it.
Reading research papers gives you much more bang for your buck in terms of developing your research interest, though is not very resume-able. But, who knows, maybe your reading will spark your own open source research project.
Be warned though, there are quite a few research papers, especially those coming out of Academia as opposed to industry, that are cryptic, lacking in detail, or not easily reproducible. And that's intentional. This isn't obvious when you read a paper, it will only become obvious when you actually try to reproduce it. These papers are the result of misaligned goals Academia in, where it's all about publishing as many papers as possible that look good on the surface. When In doubt, I'd recommend focusing on research that has a lot of interest from the industry. That's usually a safer bet.
[1] https://github.com/karpathy/convnetjs [2] https://github.com/harthur/brain
Ironically, looking dennybritz's comment, JS is particularly ill-suited for this, as, as far as I'm aware, there are no libraries for working on the GPU, which is tremendously useful for working with machine learning algorithms which usually are crunching lots of data.
However, I don't know much about karpathy's specific implementation and experiments, so there might be a reason he choose JS. I'm thinking its more likely to make visualizations easier rather than any computational reason though, so you might be stuck in a niche there.
As for Karpathy's implementation, you're right, he chose JS for that reason. His library is meant as a tool for experimenting and teaching, not to deploy these models in production.
Op: Send me an email if you're interested in this (email in profile). I actually started something similar just recently at http://www.wildml.com/.
Gitxiv is an online list combining research papers from arxiv and their implementations in github. So I would say its a great resource for what you are looking for.
Also, they run some kind of competitive implementation runs where you can collaborate with others to implement papers :
https://github.com/GitXiv/DeepQA
I am a JS dev as well interested in Deep Learning. I dont know your specific interest in this domain but I am happy to collaborate with you - if you are interested. You can reach me at raj at dhi dot io. I would suggest you to pick up python or lua. Lua is quite close to JS and Python is extremely easy to pick up as well.
There are several different types of open source projects and I'm not quite sure which kind you're looking for.
Firstly there are a lot of (often hugely popular) tools used by researchers that don't involve research directly. A good example is Jupyter (formerly known as ipython notebooks). Contributing to these projects is sure be of benefit to many people (probably more than anything else you could do), and may be a good place to learn skills because they're more likely to have good codebases written by skilled programmers. But if the project is already too large (like Jupyter), it may be a lot harder to get into.
Next up, anything that's scientific and written in javascript is most likely to be for user interface, visualisation, or just websites (PR) for research projects. For example interactive visualisations of algorithms. May or may not involve actual involvement to the research. For example I went to a seminar recently by a group of researchers here who create models for simulating the effects of policy changes for policy makers a number of government departments here. They need user interfaces for their tools, and for one project [1] apparently they're using something that generates forms on a webpage, but it's all written in Java and R, and they want to open source everything and get external contributions (the current code they have up is 2 years old). I'm not advertise them, just giving an example; there must be a number of other projects with similar needs. I don't know how you can find find them.
Finally, there may be projects that actually write the real meat of the research software directly in javascript. It sounds like this is what you want. But I don't think javascript is a popular language amongst researchers. Languages like MATLAB/Octave, Python, R, Mathematica, Magma/GAP, C, Java, Julia are mostly used. Checking my CS department's mailing list, the only mention of javascript is a project to write a Google Chrome extension to use the javascript API for an eye tracker [2]
[1]: http://www.arts.auckland.ac.nz/en/about/our-research/researc...
Much of what research is, is understanding what you're working on in the first place. Getting a solid foundation is a must, or you risk being very lost for some time, and will be fundamentally limited in what you can contribute.