Okay, forget it … it was a problem with the CA certificates on Alpine Linux. You need to run

# apk update
# apk add ca-certificates
# update-ca-certificates

before running wget, then it works :-)

FYI for others coming across this, adding the above commands to my Dockerfile did not work, I needed to specifically add the ca-certificates to wget like this:

 RUN   apk update \                                                                                                                                                                                                                        
  &&   apk add ca-certificates wget \                                                                                                                                                                                                      
  &&   update-ca-certificates    

that worked perfectly and my image built with no problems.

openssl package may be required too:

apk update && apk add ca-certificates && update-ca-certificates && apk add openssl

How difficult would it be to make an apk package for dumb-init? We're experienced in debian packaging but unfamiliar with alpine's ecosystem and would love help if it's possible

For those who arrive here, another solution is simply to use curl instead, if your idea is to run install scripts. It sounds even simpler, with less options to deal with - the default is to send to stdin for instance.

Example: to install composer on php:7-fpm-alpine I did the following:
RUN curl https://raw.githubusercontent.com/composer/getcomposer.org/master/web/installer | php -- --quiet

instead of what's suggested by Composer (that stumbles upon the CA issues on this post):
RUN wget https://raw.githubusercontent.com/composer/getcomposer.org/master/web/installer -O - -q | php -- --quiet

I had

wget: can't execute 'ssl_helper': No such file or directory
wget: error getting response: Connection reset by peer

apk add openssl solved in my case

I frown upon curl | bash installation instructions (curl | php is no different). I hope dumb-init never adopts such insecure recommendations as instructions. It grooms users with poor behaviors.

As of Alpine Linux 3.3 there exists a new --no-cache option for apk. It allows users to install packages with an index that is updated and used on-the-fly and not cached locally:

RUN apk --no-cache add openssl

Update on this;

While that fixed it for https protocol, it caused issues for http protocol. Reinstalling wget on alpine fixed wget: error getting response: Invalid argument error:

RUN apk --no-cache add openssl wget