It's not worth paying more to find a place closer to Long Street. As others have said, you'll want a car (partly because Cape Town is so spread out and geographically interrupted [lots of mountains and valleys], and partly for safety reasons).
There is a lot of street parking near Long Street and in most other areas of Cape Town that have busy nightlife. The V&A Waterfront has a cheap, well-lit parking garage, but most other places I went just had street parking. You'll encounter parking attendants who will give you a price "to stay all night"; I was really skeptical of this and had no idea whether they were officially employed attendants or just some guys looking to make a few bucks. But I think we paid something like 10 Rand for a whole evening of parking near Long St. And our (rental) car was fine.
One thing you might want to consider is what kind of parking your backpackers or hotel will provide. Some places say that they provide parking, but it's street parking (unmetered, just literally parking on the side of the street). We had to park on a steep, narrow road that made me nervous, and it was outside the gate of the backpackers' property so the car wasn't even in sight of our room. We were in a nice neighborhood, so it was fine, but it's worth accounting for. Definitely ask whether "free parking included" means street parking, parking garage, parking in a gated driveway, etc. - and whether they have a security guard.
One other note: South Africa had a lot of confusing traffic rules (for me, anyway). Almost every parking spot is marked on the pavement with a symbol, and the general road guide provided by our rental company (Avis) didn't contain all of them. I found a South Africa rules of the road website that looks pretty comprehensive, so check it out. A lot of the signs denote the type of vehicle allowed to park there or for how long / what time of day. Sometimes these rules are enforced, and sometimes they're not, but it's good to be aware of any possible consequences. :)