J’ai besoin de télécharger une application is a correct translation and will fit most uses. As for many other technical terms, English words are often used in everyday language, especially among younger speakers, so you may also hear J’ai besoin de downloader une application or, even more colloquial, J’ai besoin de downloader une app, but beware that some people may frown upon using such anglicismes. You can also find the familiar apocope appli for application.