Exporting a rasterized Row of Graphics

You can change from Row over to Grid, which does no linebreaking. Then you just need to add another set of curly braces around your former input row:

Rasterize[
 With[{size = 250}, 
  Grid[{Show[#, ImageSize -> {Automatic, size}, 
       ImagePadding -> {{30, 15}, {40, 5}}] & /@ {Plot[
       Sin[x], {x, -1, 1}], Plot[Sin[x], {x, -1, 1}]}}]], 
 ImageResolution -> 200]

Use GraphicsRow to avoid breaking the row. You can control the resolution of your exported image with the option ImageSize -> ... supplied directly to GraphicsRow. This is another advantage of GraphicsRow over Raw - it takes ImageSize option. Yet with most of other objects using plane Row is more preferable.

And though I did not use Rasterize, and exported directly, you can still use it if you want to.

Export["test.png", With[{size = 250}, 
GraphicsRow[Show[#, ImageSize->{Automatic, size},ImagePadding->{{30, 15}, {40, 5}}] & /@ 
  {Plot[Sin[x], {x, -1, 1}], Plot[Sin[x], {x, -1, 1}]},Spacings-> 0]], ImageSize -> 500]

It is possible to give an explicit ImageSize to Row and if it is large enough to contain the graphics it will not wrap. If it is given in the form {maximum} it will be sized automatically. Infinity does not appear to work so I used 1*^6:

Rasterize[
 With[{size = 250}, 
  Row[Show[#, ImageSize -> {Automatic, size}, 
      ImagePadding -> {{30, 15}, {40, 5}}] & /@ {Plot[
      Sin[x], {x, -1, 1}], Plot[Sin[x], {x, -1, 1}]}, 
    ImageSize -> {1*^6}]],
 ImageResolution -> 200]