How to do pattern matching in postgres regex?

You could try

SELECT s.name
FROM   spelers s
WHERE  s.name SIMILAR TO '(B|D)%' 
ORDER  BY s.name

I've no idea whether or not either the above or your original expression are sargable in Postgres regex though.

If you create the suggested index would also be interested to hear how this compares with the other options.

SELECT name
FROM   spelers
WHERE  name >= 'B' AND name < 'C'
UNION ALL
SELECT name
FROM   spelers
WHERE  name >= 'D' AND name < 'E'
ORDER  BY name