I built a ServiceConnect paclet for this (details on how I did this can be found at the end of this post).

You can access it like so:

PacletInstall[
 "ServiceConnection_NASA",
 "Site"->
  "http://www.wolframcloud.com/objects/b3m2a1.paclets/PacletServer"
 ]

(note that unless you have BTools installed and have loaded the PacletTools package (e.g. by just calling PacletUpload raw to get the autoloader to run) you won't get this formatted form)

Then just connect and have at it!

$so = ServiceConnect["NASA"]

Pic of the Day

$so["PictureOfTheDay"]["Image"]

Earth

$so["EarthImages",
  "Date" -> Today - Quantity[1, "Weeks"], 
  "ImportImages" -> True,
  "ImportSelection" -> ;; 3
  ][All, {"Image", "caption"}]

Mars

$so["MarsImages", 
  "earth_date" -> Today, "ImportImages" -> True, 
  "ImportSelection" -> {1}
  ][2]

And more!

You can get the currently supported requests like so:

$so["Requests"]

{"APIKey", "Authentication", "EarthImages", "ID", \
"ImportEarthImages", "Information", "LandsatAssets", "LandsatImages", \
"MarsImage", "MarsRoverData", "Name", "NeoFeed", "NeoLookup", \
"Patents", "PictureOfTheDay", "RawRequests", "RequestData", \
"RequestParameters", "RoverCameras"}

And you can figure out what to pass to each using the "RequestParameters" like so:

$so["RequestParameters", "Request" -> "Patents"]

<|"Parameters" -> {"api_key", "query", "concept_tags", "limit"}, 
 "Required" -> {}|>

$so["Patents"]["results", 1, "title"]

"Selective functionalization of carbon nanotubes based upon distance  \
   traveled"

If you want me to implement more requests I can easily do so

Construction Process

I built this according to the process laid out in my answer to the question Creating a custom web API ServiceConnection.

The actual construction notebook can be obtained here so you can see what went into building it.

Essentially, I just have a template notebook that you fill out and which formats a call to the CustomServiceConnection function in my BTools package.