Why has communism had such a lasting influence on human rights in China?
The reason for this is that communism is a system of government that does not believe that individual rights matter to any great degree. Instead, it holds that the needs of the society are more important than individual rights.
By communist theory, there needs to be a "vanguard party" that will promote the cause of the proletariat and will fight against any counterrevolutionary activity. To communists, any activity that might work against that vanguard party is counterrevolutionary and therefore bad for society.
This is why China's government is so willing to abridge the human rights of its people. It feels that it is the vanguard party that is protecting the rights of the proletariat. Therefore, anyone who opposes it (whether it be the Falun Gong or dissidents or whoever) is working to hurt the country as a whole. Such people must be suppressed where, in a democratic system, they would be allowed to speak and act as they wish.