Assuming that your server expects that the username:password
combo will be encode it UTF-8 (see RFC 7617 for more details) then use this:
import 'dart:convert';
import 'package:http/http.dart';
main() async {
String username="test";
String password = '123£';
String basicAuth="Basic " + base64.encode(utf8.encode('$username:$password'));
print(basicAuth);
Response r = await get(Uri.parse('https://api.somewhere.io'),
headers: <String, String>{'authorization': basicAuth});
print(r.statusCode);
print(r.body);
}