Is a person held accountable for the sin of his female relative using a program he installed for her to download songs and useless applications, knowing that his intention was for general benefit and he did not anticipate the negative use?

There is no sin upon you merely because your relative uses this program to download unbeneficial songs. However, if the songs are forbidden, then you must advise her gently and wisely, and guide her to utilize her time in what benefits her in matters of her religion and worldly life. For it has been authentically narrated that for Allah to guide a single man through you is better for you than red camels.