My mother's date of birth in all documents is 28/03/1973. But in the passport it is 01/01/1973 as this is the date of birth in her matriculation certificate.
Will this cause a problem if she travels abroad? What should be done?
It should not be a problem. Only the passport is normally shown during travel.
Does the passport date match her Indian national ID card date of birth? That's the only other thing I can think of that might get compared during travel.
For visa applications, bank statements are commonly required. Do Indian banks print the date of birth of the account holder on the statement? (My guess is no).