It used to be pronounced vuh-GARE-i, but it's changed. As @Tim above stated:
(GenAm) IPA: formerly /ˌvəˈɡɛɹi/, now commonly /ˈveɪˌɡəɹi/
It's usually pronounced VAY -guh-ri, but there are still people who pronounce it the "other way", as can be seen here
The /ˈɛryən/ pronunciation is just a result of English phonology processing a foreign borrowing that starts with the letters AR.
Aryan is a borrowed word in all languages outside the Indo-Iranian subfamily of Indo-European. The rest of the world pronounces it as some variant of [arjan], which comes, as noted, from Sanskrit ārya /a:ryə/ 'compatriot'. Therefore, /'aryən/ is a perfectly acceptable English pronunciation, and the only acceptable one when using the term in its modern Indian sense.
Any use of Aryan (outside scare quotes) that refers to Germany or white racism is a result of romantic interpretations of 19th century German linguistic scholarship (e.g, Grimm's Law), which unearthed the prehistory of the "Indo-Germanic" (as I-E was then called, from names of its Eastern- and Westernmost families) languages. It was all very exciting, apparently. See also Wagner, Mad King Ludwig, Neuschwanstein, German Empire.
The AHD of IER says that Skt ārya comes from the PIE root *aryo- 'Self-designation of the Indo-Iranians'; other descendants of the same root are Iran and, surprisingly, Eire -- Celtic languages sometimes retain PIE roots that are otherwise lost in the Centum group.
None of these are English words, and so English treats them the same way it treats all borrowed words -- it changes the pronunciation until it tastes right. That's all.
Edit:
I almost forgot, another reason to pronounce Aryan /'aryən/ is because Arian /'ɛriən/ usually refers to Arianism, a very important variety of Christianity that was the religion of the Ostrogoths in Italy, and the Visigoths in Spain. The only Gothic texts known to exist are translations of various parts of the (Arian Christian) New Testament.
Best Answer
Essentially the same question was also posted on the linguistics stackexchange, several months after the posting of the present question here on the English stackexchange. As of Jan 14, 2016, here is what we have learned there:
At the present time, there does seem to exist a subgroup of adult native English speakers who pronounce the "th" as voiced. It is not clear if there is any underlying reason for this that would be common to all or even most of them. There do seem to exist some regional tendencies (see 2. below), but it seems there must be some other factors at play as well (see 3. below).
There indeed exists some published research on this: the Survey of English Dialects (Orton et al. 1962-71) recorded "a number of instances of the voicing of all four fricatives in word-initial (Anlaut) position in the South and the South-West Midlands" (Fisiak 1988); see Voitl 1988 (also here). Related research includes that by Martyn Wakelin, esp. Wakelin and Barry (1968) (thanks to Alex B. for all of this).
From the comments by musicallinguist, it seems that sometimes even among siblings raised in the same household (of native English speakers, in an English-speaking country) there can be a difference in the voicing of the initial "th," which then persists into adulthood. References to published research on this aspect would be most welcome.