Who is qualified to understand Vedas

There is a stricture that no one should try to learn the Vedas 
who is not a qualified Brahmana. This stricture has been 
wrongly interpreted in so many ways. A class of men, who 
claim brahmanical qualification simply by their birth right in 
the family of a Brahmana, claim that the study of the Vedas 
is monopoly of the brahmanical caste only. Another section 
of the people take this as an injustice to other castes who do 
not happen to take birth in a brahmanical family. But both of 
them are misguided. The Vedas are subjects which had to 
be explained even to Brahmaji by the Supreme Lord.

Therefore the subject matter is understood by persons with exceptional qualities of goodness. SB 1.4.24.

Bhagavad Gita is the essence of all Vedas. It can be 

understood by a person who is friend and devotee of Lord 
Krsna, by approaching a bona fide Acharya (Guru) in a 
humble manner.