Modern Hebrew is not SAE by any stretch. Going through Haspelmath's criteria:
Definite and indefinite articles: Modern Hebrew (MH) has only a definite article (-ה), which is inherited from Biblical Hebrew (BH)
Relative clauses formed by a relative pronoun: MH's relative clauses are formed by the particle ש- and not by a pronoun (inherited from BH)
"Have"-perfect: MH has no have perfect, nor does it have a way of distinguishing perfect from preterite (past-future tense is inherited from BH's perfect-imperfect aspect)
Nominative experiencer: In some cases MH is like SAE (such as אני שמח "I am happy"), in other cases not (such as the dative subject in חם לי "I am hot"); note that while this is not SAE, these two ways exist in some European languages which are nevertheless considered SAE, and Haspelmath uses the arbitrary division of a ratio of < 0.8 to define SAE. (Following Amir Zeldes' analysis from Aharon M. Vertmont's answer, MH does have a tendency towards the nominative experiencer, but that the same is true for BH.)
Participial passive: Passive construction is a separate conjugation, inherited from BH
Anticausative prominence: Since Haspelmath's article measures this by percentage of lexicon I can't judge this one, but Hebrew has a separate conjugation for causative verbs which suggests it might meet this criterion. (Following Amir Zeldes' analysis from Aharon M. Vertmont's answer once more, MH is in fact anticausative, and this is likely the same as BH.)
Dative external possessors: This does exist in MH (e.g. לקח לי "he took from me") and is not inherited from BH
Negative pronouns and lack of verbal negation: MH has the non-SAE NV + NI type described in the article (e.g. אף אחד לא בא "no one comes," but literally "no one doesn't come")
Particles in comparative constructions: The comparative is מ- which means "from," inherited from BH
Relative-based equative constructions: This doesn't exist in MH, which uses "X like Y" instead of the SAE "as X as Y" and is inherited from BH
Subject person affixes as strict agreement markers: This true in MH only in the present tense, whereas the past and future tenses are (like BH) pro-drop
Intensifier-reflexive differentiation: Reflexive pronouns such as עצמי "myself" (inherited from Mishnaic Hebrew) are not differentiated from intensifier pronouns
In nearly all of these cases Modern Hebrew is unambiguously not SAE. If we follow the numerical analyses from Zelde's article, only features #4, #6 and #7 are SAE, for a total of only 3 SAE features. More importantly, the SAE features in #4 and #6 are also present in Biblical Hebrew, leaving only one feature, #7 as the product of SAE influence.