Yes a landlord is responsible for plumbing issues and other things such as electrical problems heating problems etc.
A landlord by law has to maintain the home or apartment and keep the home or apartment up to code and provide safe and habitable living conditions for the tenants.
So yes if you have a problem with your plumbing you should get into contact with your landlord and have them repair the problem.
If they don't fix the problem or get the plumbing issue fixed in a reasonable time frame then you can take the landlord to court over the problems.