I think other editors make very valid points above: speaking with an accent unfortunately can collide with pre-existing stereotypes that we have, which encourage hearers to make judgements on that person based on the accent. As much as we deplore this we do know that this is a fact and have to recognise it. I do think though that things are gradually changing thanks to globalisation and accents are being more and more widely accepted.
I personally love accents of all types, but that's just me. It seems like you might be focusing specifically on Asian accents for a reason though. In general, it seems that an older person with an Asian accent is perceived as "traditional" or "old school". A younger person with an Asian accent is perceived as "first generation" or "from another country". Because Asian people are generally stereotyped as being intelligent and hardworking, the accent does not seem to leave a negative impression at all. Of course, this does not mean that stereotyping is OK.
This is only based on my observations of high school and college students. I certainly cannot claim to be an authority!
I like both points a lot.
I think in educated circles with exposure to the world, accents do not leave a bad impression or less of a bad impression. Only when people are very ethnocentric and narrow in their thinking, do they really mind. It stems from ignorance and insecurity, in my opinion.
With that said, in the real word, speaking well is important. How you dress is important. Unfortunately, we live in a superficial world.
Of course to most people someone who speaks with an accent makes somewhat of a bad impression. But it's not something that can't be overcome. My father has something of a Filipino accent even after being here since 1960 and it does hurt him in terms of being taken seriously. But while it did hurt at times, it didn't keep him from having a good career. No matter what things should be like, I think it's inevitable (and not an American problem only) that people who speak any language with an accent will not be taken as seriously as those who speak properly.
One thing Americans should keep in mind is that everybody who speaks English with a foreign accent also speaks at least one more language than the average American, and fluently at that!
Far worse than an accent is the use of non-standard English or poor diction. If a person of Asian or other origins can speak without so strong an accent that he/she can be understood, many people find the difference enchanting.
When a person works hard enough to speak fluent English, the effort does not go unappreciated, especially nowadays with so many who literally refuse to learn English.