use warnings;
use strict;
use Data::Dumper;

my (%result, $key, $val);

while (<DATA>) {
    if (/^>(.*)$/) {
        $result{$key} = $val if $val;
        
        $key = $1;
        $val = '';
        next;
    }

    chomp;
    $val .= $_;
}
continue {
    $result{$key} = $val if $val;
}

print Dumper \%result;

__DATA__
>SEQ1
-----I--RL--AAIDVDG-NLT----------D--R--D-RL-ISTKA-IESIRS--A-
-E-K--------K-GLT-VSL----LS------GN-V----I-PVV---YA-L------K
IF---------------L-----GINGPVF------------------------------
>SEQ2
-MKI----KA--ISIDIDG-TIT------YPN-R-------MIHEK--A-LEAIRR--A-
-E-S--------L-GIP-IML----VT------GN-T----V-QFA---EA-A------S
IL---------------I-----G----TS-----------------GP-VV--------
>SEQ3
--KI----KA--ISIDIDG-TIT------YPN-R-------MIHEK--A-LEAIRR--A-
-E-S--------L-GIP-IML----VT------GN-T----V-QFA---EA-A------S
IL---------------I-----G----TS-----------------GP-VV--------
---AE--D------GG---A---------------------------------------I
[download]

$VAR1 = {
          'SEQ3' => '--KI----KA--ISIDIDG-TIT------YPN-R-------MIHEK--A
+-LEAIRR--A--E-S--------L-GIP-IML----VT------GN-T----V-QFA---EA-A-----
+-SIL---------------I-----G----TS-----------------GP-VV-----------AE--
+D------GG---A---------------------------------------I',
          'SEQ2' => '-MKI----KA--ISIDIDG-TIT------YPN-R-------MIHEK--A
+-LEAIRR--A--E-S--------L-GIP-IML----VT------GN-T----V-QFA---EA-A-----
+-SIL---------------I-----G----TS-----------------GP-VV--------',
          'SEQ1' => '-----I--RL--AAIDVDG-NLT----------D--R--D-RL-ISTKA
+-IESIRS--A--E-K--------K-GLT-VSL----LS------GN-V----I-PVV---YA-L-----
+-KIF---------------L-----GINGPVF------------------------------'
        };
[download]

I thought initially I could just split on new line, but I cant due to the sequence data occurring over several lines

Split on entire records, a record being header + sequence.  It looks like the appropriate record separator would be newline followed by '>' (i.e. "\n>"). Then split each record into header and sequence, by treating stuff up to the first newline as header.