I think cousin marriage should be banned, particularly where it is practised repeatedly across generations.
PPs have pointed out that marriages between relatives occurred throughout European history, including among aristocratic and royal families. That's true. But there is a distinction between occasional marriages between relatives for dynastic, political or financial reasons and a cultural pattern where close relative marriages are repeated within the same extended families over multiple generations.
First cousins share more genetic material than unrelated couples, increasing the chance that both parents carry the same recessive genetic condition. Where cousin marriage is repeated generation after generation within a relatively closed gene pool, those risks can become more significant.
There is also a problem with describing this simply as an "Asian" issue. Asia is enormously diverse, and cousin marriage is certainly not practised universally. For example, in Hindu families of Indian origin, first-cousin marriage is considered completely unacceptable (even the thought of it is outrageous) because first cousins are regarded, essentially, as brothers and sisters (and second cousins as nieces and nephews).
The difficulty is about how a ban can actually be enforced. A law would need to apply universally, regardless of ethnicity or religion (a distant great uncle and great aunt on DH's side are first cousins. Both white British. No one likes to talk about it!).
Alongside any legislation, there would need to be much greater education about the genetic risks of repeated consanguineous marriage so families can make genuinely informed decisions about marriage and having children. But, then, how this can be enforced for marriages that have taken place abroad or as religious ceremonies? Would we be taking away more legal rights from the women who might be forced/coerced into it?