What are the functions of a nuclear membrane?

The nuclear membrane performs two major functions that are critical to the survival of the cell. 

It acts as a barrier between the nucleus and cytoplasm - The primary function of the nuclear membrane is it acts as a barrier that protects the contents of the nucleus, including the genetic materials, from the cytoplasm. Without the nuclear membrane, the cytoplasmic contents would enter the nucleus and mix with its contents, destroying the DNA. This would hamper the functioning of the cell, eventually resulting in cell death. 

It regulates the materials that enter and leave the nucleus - The selectively permeable nature of the membrane and the nuclear pores play a pivotal role in regulating the materials that enter and exit the nucleus. Only small nonpolar molecules such as mRNA and proteins can pass between the nucleoplasm and the cytoplasm.