some quotes from dharma saastras.

आपो नारा इति प्रोक्ताः आपो वै नरसूनवः।
ताः यदस्यायतनं पूर्वं तेन नारायणः स्मृतः॥
ब्राह्मण्यां ब्राह्मणात् जातः संस्कृतो ब्राह्मणो भवेत्।
एवम् क्षत्रिय वित् शूद्रः ज्ञेयाः स्वेभ्यः स्वयोनिजाः॥
अध्यापनं च अध्ययनं यजनं याजनं तथा
... दानं प्रतिग्रहश्चैव षट्कर्माण्यग्रजन्मनः॥
तपो विशेषैः विविधैः व्रतैश्च विधि चोदितैः
वेदः कृत्स्नोऽधिगन्तव्यः सरहस्यो द्विजन्मना॥
aapo naaraa iti proktaaH aapo vai narasuunavaH.
taaH yadasyaayatanaM puurvaM tena naaraayaNaH smR^itaH..
braahmaNyaaM braahmaNaat jaataH saMskR^ito braahmaNo bhavet.
evam kSatriya vit shuudraH j~neyaaH swebhyaH swayonijaaH..
adhyaapanaM cha adhyayanaM yajanaM yaajanaM tathaa
daanaM pratigrahashchaiva SaTkarmANyagrajanmanaH..
tapo viesheSaiH vividhaiH vrataishcha vidhi choditaiH
veadaH kR^itsno.adhigantavyaH sarahasyo dwijanmanaa..
waters are called naaraas because the waters are the children of Nara an incarnation of Vishnu.. That Vishnu who made his resting place the Naaraas or waters is called Narayana.

A child born to a brahmin woman on a brahmin male and is purified by the necessary samskaaraas is brahmana. Kshatriya, Vaisya and Sudra are also determined by the yonis in which they are born.

Teaching, studentship, doing yajnas, performing yajnas on behalf of others, giving gifts and receiving gifts are the six functions of a brahmin.

The Vedas are to be learned and practices by brahmin after learning it through strict observation of penances, rituals, fastings, all prescribed by shastraas, and such 

Vedaas are to be learned with precision and  utomost care to maintain its secret nature by the Dwijas.

These are some quotations from traditional dharmasaastras.. to accept  those beliefs or not is left to  the choice of the learned and the discerning people ..