The Democrats are MUCH MORE effective in reaching Asians. They are 5% of the public and (generally) have a lot more money. While they should be Republicans, the Democrats hit very hard in the immigration area, which is a big deal to Asians trying to get their families over here.
It’s a tough hurdle to overcome for Republicans to overcome, it wouldn’t hurt if they tried a bit...such as finding a conservative or two that speaks Mandarin, Korean, or Vietnamese...and explaining that our problem with immigration is not with Asian immigrants bringing their families in legally (within reason - no third cousins please), but rather with the ones sneaking over from the South.
How about Bobby Jindal? I think you’ll see Asians increasingly turn to the Republicans as they understand more thoroughly how the tax code works; as a generally hard-working people, they have nothing but contempt for the permanent underclass they are expected to provide for.