We know that Drift velocity
$v _{ d }=\frac{ eE }{ m } \tau=\frac{ e }{ m } \tau\left(\frac{ V }{1}\right)$
$\left(\because E =\frac{ V }{1}\right)$
so for a particular conductor of a particular length the drift velocity will directly depend upon voltage
Hence $v_{d} \propto V$.