Background: I work at an independently-owned YHA hostel in Australia. YHA is a non-profit organization that owns many but not all YHA-branded hostels in Australia. YHA is the Australian branch of Hostelling International, HI. The hostel I work at is not non-profit, just a member of the network conforming to its standards.
Anyway, for my hostel organization there are three models of online booking systems as far as I can make out. I work at reception, not in the back office.
Website takes a 10% or 12% payment from the customer. Customer must pay the balance directly to us when they arrive. Booking.com, HostelBookers, and HostelWorld, are examples of sites that use this system. You only pay this "deposit" in advance.
Website takes full payment and either charges 10% or 12% above our rates, or gets 10% or 12% of the price we advertise, with us getting the remainder. Payment is settled between the website and hostel by invoice. Agoda is an example of a site that uses this system. You pay in full in advance.
Website belongs to the hostel or organization the hostel is a part of. Full advertised price is charged to the customer and goes to the hostel or organization. YHA is an example using this system. I believe HI uses such a system internationally too, but I'm less familiar with it. You pay in full in advance.
Then of course we also take bookings not via any "system". This is done via e-mail, telephone, or in person. Our policy is that you just pay the full price of your first night as your deposit when booking. The Christmas/New Year period is special and must be paid in full when booking. Of course if you ask to pay in full when you book we do do that as well.
This is just an example from my experience meant to illustrate some of the possibilities.
So yes it is possible to pay in advance for hostel booking. But it may depend on various factors.
I would always recommend using the hostel's native booking website, if it has one. Many independent hostels not part of a bigger organization don't have one. Also, it's not unheard-of that a hostel will have a special price available only via one third party booking site. My hostel was doing so at the time I left for my trip.
If there's not a hostel/organization booking site, try Agoda. I am sure there must be some other online booking sites that use a similar model to Agoda.
Or email, or call by phone or Skype. If it's international, do check the timezone first. Be aware of the international date line. Have a Visa card or Mastercard ready.
Some independent hostels do not make it easy to find their phone number or email address because it's easier to rely on the booking sites doing that for them and many booking sites want you to book through them so don't post the phone and email details of their member hostels since you could circumvent their fee after using their service to find the accommodation.