New Jersey Governor Chris Christie made headlines across the nation when he decided unilaterally to cancel a Hudson River rail transit tunnel to Manhattan after estimated costs ballooned from $8.7 billion to $14 billion. Wendell Cox has a useful summary of the main issues over at NewGeography.com and a spirited defence of Governor Christie's decision; it's short and well worth a read.
Nevertheless, a potenial resolution to the cost concerns may be addressed if the tunnel is proposed as part of a long-term concession to a private firm through a public-private partnership. These projects are common outside the US (I discuss them in my books Mobility First and The Road More Traveled as well as on the Reason Foundation web site), and they have substantial benefits if structured properly.
The key is to capture the private-sector expertise in building and managing these facilities while minimizing the potential risk to taxpayers. New Jersey and New York, working with the U.S. Department of Transportation, should invite bids from private companies to build the tunnel. The state and federal governmens (perhaps through the Port Authority) would guarantee a steady revenue stream through what are called "availability" payments based on the best bid from the private sector. This would lock-in public sector revenue streams while shifting the risk for cost overruns to the private sector.
A variety of ways exist to structre this kind of a public-private parternship and details can be found at the National Council for Public Private Partnerships web site (including an summary of financing and construction arrangements).