Map[{FromDigits @ #, {##2}} & @@@ # & @* StringSplit] @ StringSplit[l, "\n"]

{{{1, {"atattaggtt", "tttacctacc", "caggaaaagc", "caaccaacct"}}, 

  {61, {"ctctaaacga", "actttaaaat", "ctgtgtagct", "gtcgctcggc"}},
  {121, {"gcagtataaa", "caataataaa"}}}}

Also

f = Normal[{#, DeleteCases[{##2}, ""]} & @@@ SemanticImportString[#]] &;
f /@ l

{{{1, {"atattaggtt", "tttacctacc", "caggaaaagc", "caaccaacct"}}, 

  {61, {"ctctaaacga", "actttaaaat", "ctgtgtagct", "gtcgctcggc"}}, 
  {121, {"gcagtataaa", "caataataaa"}}}}

Map[{FromDigits @ #, {##2}} & @@@ # & @* StringSplit] @ StringSplit[l, "\n"]

{{{1, {"atattaggtt", "tttacctacc", "caggaaaagc", "caaccaacct"}}, 

  {61, {"ctctaaacga", "actttaaaat", "ctgtgtagct", "gtcgctcggc"}},
  {121, {"gcagtataaa", "caataataaa"}}}}

Also

f = Normal[{#, DeleteCases[{##2}, ""]} & @@@ SemanticImportString[#]] &;
f /@ l

{{{1, {"atattaggtt", "tttacctacc", "caggaaaagc", "caaccaacct"}}, 

  {61, {"ctctaaacga", "actttaaaat", "ctgtgtagct", "gtcgctcggc"}}, 
  {121, {"gcagtataaa", "caataataaa"}}}}

and

f2 = {FromDigits @ #, {##2}} & @@@ 
      Split[StringSplit @ #, Not @* StringMatchQ[NumberString] @* First] &;
f2 /@ l

 {{{1, {"atattaggtt", "tttacctacc", "caggaaaagc", "caaccaacct"}},

   {61, {"ctctaaacga", "actttaaaat", "ctgtgtagct", "gtcgctcggc"}},
   {121, {"gcagtataaa", "caataataaa"}}}}