Yes you should always wash your hands after blowing your nose to prevent the spread of germs.
Sometimes you may be out somewhere and have to blow your nose and not be able to wash your hands but you should avoid touching any food before washing your hands.
Always wash your hands after blowing your nose as soon as possible using warm water and antibacterial soap.
Wash your hands after blowing your nose and coughing as well.
Keeping your hands washed prevents the spread of germs and your boogers and snot are full of germs that can easily be spread to other people.
So use a tissue too blow your nose and then wash your hands right after you blow your nose if possible.