Fast/efficient way to turn real passband samples to complex baseband
In the particular case of BPSK transmitted over wireline, there is no quadrature component, so the equivalent low-pass representation (the complex envelope) is real. You can get the complex envelope simply by downcoverting the signal from the carrier frequency down to DC.

For general baseband signals, or over the wireless channel, the process can be outlined as follows:


Find the analytic signal. Say the passband signal is $r(t)$. Then, $r_+(t)=r(t)+j\hat{r}(t)$ has no negative frequencies ($\hat{r}(t)$ is the Hilbert transform of $r(t)$).
Downcovert $r_+(t)$. The signal $x(t)=r_+(t)e^{-j2\pi f_c t}$ is the complex envelope.


Note that implementing this algorithm is not necessarily trivial. In practice, you almost always want to downconvert $r(t)$ to some intermediate frequency before attempting the Hilbert transform.

An alternative procedure is as follows:


Center the positive part of the spectrum around DC, by calculating $y(t)=r(t)e^{-j2\pi f_c t}$. Note that the resulting signal is complex even in the BPSK case.
Get rid of the negative part of the spectrum by low-pass filtering $y(t)$.


The downside of the second procedure is that it doubles the maximum frequency of the signal. Mathematically both procedures are identical (up to a constant amplitude factor).