<Page>





                                                                    Exhibit 32.1


                            CERTIFICATION PURSUANT TO
                 18 U.S.C. SECTION 1350, AS ADOPTED PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002


In connection with the Quarterly Report on Form 10-Q/A of Globix Corporation
(the "Company") for the period ending March 31, 2002 as filed with the
Securities and Exchange Commission on the date hereof (the "Report"), I, Peter
K. Stevenson, Chief Executive Officer of the Company, certify, pursuant to
U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley
Act of 2002, that:

The Report fully complies with the requirements of section 13(a) or 15(d) of the
Securities Exchange Act of 1934; and

The information contained in the Report fairly presents, in all material
respects, the financial condition and results of operations of the Company as of
March 31, 2002 (the last date of the period covered by the Report).



                                             /s/ Peter K. Stevenson
                                             Peter K. Stevenson
                                             Chief Executive Officer