John Hare has an article about commercial beyond Earth orbit exploration in a world where cheap access to space has lowered the cost of a kg to LEO to $1000 or less. But I think it begs the question, how much does it cost now?
Warning: contains Machiavellian humor.
The biggest problem with commercial human beyond LEO flight right now is the lack of an affordable LH2 upper stage. In SpaceX terms: they don't have Raptor yet. But hey, no-one ever said you have to use LH2/LOX stages to go beyond LEO.
If you're aiming at an asteroid at an optimal time, there's at least one target you can hit with only 2.8km/s of delta-v from LEO, and 1km/s of delta-v to rendezvous with the asteroid, 2006 RH120. (Note that you can divide this up any way that makes sense to you, but more than 1km/s of delta-v at rendezvous is probably undesirable. For a flyby of 2006 RH120 you need 3.733km/s). See this list. Personally, I'd rather aim at 2009 BD as it is almost always "close" to the Earth.
Now the vehicle. Let's say a Dragon with a full service module, crew of 2 or 3 and some supplies: that's 10t. Even in the ISS servicing configuration it has sufficient delta-v to do the 1km/s delta-v rendezvous, and do the 1km/s delta-v needed to get back. Slowing down when you get to Earth orbit will be a dicey situation, but if you can burn off some delta-v with a lunar flyby then the PICA heatshield on the Dragon should be sufficient to get us safely back on the ground. But hey, who said anything about coming back anyway.
Using Kirk Sorensen's great formulas, the initial mass in LEO will be less than 27t (for anyone who cares, I'm using lambda=0.0133707, phi=0.0216, ISP=342s). This means you need about 17t of fuel, minus tankage, and that's not so bad. First flight will be the Falcon 9 to deliver 10t of LOX in an insulated tank payload. Depending on how much boil-off there is before the next flight, and taking into account the mixture ratio, the next flight will carry 5200kg of RP-1 and 4800kg of LOX in payload. Both the upper stages will remain SpaceX's, so they can reuse it or whatever they think they can do with it.
Finally the manned Dragon is launched to join up with the propellant tanks. Unlike the other two flights, ownership of the upper stage is transferred from SpaceX to us, as is the Dragon, but I hear they're prettimuch assuming that model for GTO flights, so it shouldn't be a problem. And they can have the Dragon back if they really want our smoking corpses, uhh, I mean, ya know.
The entire stack heads off to the asteroid. Note that it's the upper stage of the manned Dragon flight that is acting as the trans-asteroid-injection stage. Depending on how close the target is, it's a few month voyage. If you want to get fancy you can take along a really long tether and swing up the vehicle to get artificial gravity.. but remember that it'll eat into the Dragon's propellant mass, reducing contingency on rendezvous.
It's quite an adventure, so what's the price tag? The two tanking flights are just stock Falcon 9 flights at $56M each. The manned flight is probably going to be something like $150M. The tank hardware with its insulation and such is in the noise. All up, $265M.
That's a lot of money to go asking a venture capitalist for. You're going to need a pretty fancy view graph presentation to convince them that you can pull it off without dying horribly, do it better than a 1t robotic probe that could be sent direct on a single Falcon 9 flight, and bring back something worth at least 10 times as much as it cost to execute [the crew, no really, just kidding].