Background checks serve many purposes. One is the criminal history, about which you write. Another purpose is to see if the prospective tenant pays his or her rent and other financial obligations. One way to provide extra security for the rent obligation is for you to co-sign the lease or to guarantee the lease (if all else fails -- agreeing to co-sign or guarantee is not good for you, and may negatively affect your credit availability).
An apartment complex is supposed to list the selection criteria for prospective tenants, which can include criminal history. See Texas Property Code section 92.3515, which you may review at the following web address:
Considering her health issues, a landlord may not give much consideration to 20 year old criminal matters. But, do not falsify the application, as that alone is grounds for rejection or a later eviction. You may wish to take a personal role in the application process to paint your grandmother's situation in the best and most sympathetic light, which will probably give her a much better chance of securing an apartment.
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