If I claim unemployment benefits, will it impact my sponsors and will they get problems from government?

You can receive unemployment benefits without consequences for your sponsors. Your immigration sponsors promised the U.S. government that they will take care of your living and health care costs if you are unable to do it yourself, and that you will not apply for welfare, food stamps, Medicaid, SSI, subsidized housing, and other forms of public support. Unemployment benefits are a form of insurance; premiums for this insurance were paid by your employer; so, technically, payouts under this insurance do not make you a "public burden" Don't worry, if you apply for a benefit to which you are not entitled as a sponsored immigrant, your application will be denied.