Is there a cross browser way(s) of detecting support for asian languages, or japanese specifically?
Do you just mean “is there a Japanese font installed”? Because the only other aspect of “support for Asian languages” that modern OSs can have configured is to do with whether character encodings like Shift-JIS are known (aka code page 932 on Windows). But you don't generally need that anyway since you can just use UTF-8 which is much cleaner and supported everywhere.
In general you can't directly tell whether a font for a particular character range is installed or not, but there are roundabout ways. Like for example measuring the rendered size of on-page text:
var half= document.createElement('span');
var full= document.createElement('span');
half.appendChild(document.createTextNode('\uFF71'));
full.appendChild(document.createTextNode('\u30A2'));
document.body.insertBefore(half, document.body.firstChild);
document.body.insertBefore(full, document.body.firstChild);
var havejapanesefont= half.offsetWidth!=full.offsetWidth;
document.body.removeChild(half);
document.body.removeChild(full);
This temporarily creates elements containing half-width and full-width variants of the katakana ‘a’. If a Japanese font is available we would expect them to be different widths. If no such font exists, they will most likely be rendered as similar ‘unrenderable character’ glyphs with the same width.