Whatever the question is, jQuery Mobile is never the right answer. Well, unless you’re throwing something together very quickly and have absolutely zero concern for customisability and are targeting only phones with screens 4.5” and smaller.
Why? Because any phone with a screen bigger than say 5” the website will just look silly. To whatever question might prompt the response “jQuery Mobile” the correct answer will always be ”Responsive website.” If you design a website to be responsive, you limit yourself to nothing. From the smallest screen to the largest you cover them all, leaving no gaps. Take this very website for example - granted I did cheat (only because I have zero design skills) and purchased a responsive template - but make the window big, then slowly shrink it down and watch how the website lays itself out to fit whatever the size of the window.
This makes a whole lot more sense than saying “I need to build a mobile website” and taking off down the path of building a mobile-targeted website. Then you realise half way (or further) through planning there are questions like: “What browsers will we support? What resolutions should we make the website for? What devices are we targeting?” If you make a responsive website these questions become redundant. Take a leaf out of 320 and up’s book and design for the small screen, then work your way up. Do that, or use Twitter’s Bootstrap to base your website on. You’ll thank me later.