The United States Department of Transportation was formed in 1966 and began operations in 1967 in the Johnson administration to consolidate a set of federal agencies dealing with transport, which had been in various places. One of the most important components was the Federal Highway Administration, which Wikipedia notes:

The organization has several predecessor organizations and a complicated history. The Office of Road Inquiry (ORI) was founded in 1893. In 1905 that organization’s name was changed to the Office of Public Roads (OPR) which became a division of the United States Department of Agriculture. The name was changed again to the Bureau of Public Roads in 1915 and to the Public Roads Administration (PRA) in 1939. It was then shifted to the Federal Works Agency which was abolished in 1949 when its name reverted to Bureau of Public Roads under the Department of Commerce.

The Urban Mass Transit Administration, founded in 1964 was in HUD (and even a few years after USDOT was formed).

The National Aeronautical and Space Administration was founded in 1958 by incorporation of the National Advisory Committee for Aeronautics, itself formed in 1915!

Today USDOT has the following units:

• Federal Aviation Administration (FAA)
• Federal Highway Administration (FHWA)
• Federal Motor Carrier Safety Administration (FMCSA)
• Federal Railroad Administration (FRA)
• Federal Transit Administration (FTA)
• Maritime Administration (MARAD)
• National Highway Traffic Safety Administration (NHTSA)
• Office of Inspector General (OIG)
• Office of the Secretary of Transportation (OST)
• Pipeline and Hazardous Materials Safety Administration (PHMSA)
• Research and Innovative Technology Administration (RITA)
• Saint Lawrence Seaway Development Corporation (SLSDC)
• Surface Transportation Board (STB)

Two former agencies, Transportation Security and the Coast Guard, were moved to Homeland Security.

USDOT’s mission is to “Serve the United States by ensuring a fast, safe, efficient, accessible, and convenient transportation system that meets our vital national interests and enhances the quality of life of the American people, today and into the future.”

NASA’s Vision is “To reach for new heights and reveal the unknown so that what we do and learn will benefit all humankind.”

So the scopes are a bit different, USDOT cares about Americans, while NASA cares about humanity. NASA is about exploring the unknown and USDOT is about the exploiting the known. But without a sufficient dialog between the two, the unknown never becomes known. Research and practice need to be integrated for progress to be made.

Some Reasons NASA should be in USDOT

1. It will decrease the myopia of USDOT, encouraging that organization to have a longer term and far broader perspective on transportation, now and in the future.
2. It will install in USDOT a sense of the future possible
3. It will increase coordination between NASA and mainstream transportation practice (especially, but not only aviation), tying research more closely to implementation.

Some Reasons NASA should not be in USDOT

1. It will diminish NASA’s independence.
2. NASA will have to compete more directly with today’s needs for federal funding and attention. The 2012 NASA budget was $17.7 B, while USDOT was $72.6 B. So NASA would be the second largest agency in USDOT, following only FHWA, but ahead of FAA and FTA.
3. Government agencies should be smaller and more focused. USDOT will lose focus with NASA on board, but more importantly, NASA will lose focus responding to USDOT issues.
4. The skills and missions of the  agencies are very different, and irreconcilable. The short term is the enemy of the long term.