This is a very interesting sentence, as it allows me to show the difference between formal and informal Italian.
I never heard this sentence on our public transports, so this is completely made up, but it certainly resembles what one expects to hear.
Italiano - Formal Italian used for broad communications
Si pregano i gentili passeggeri di afferrare saldamente un elemento dell'autobus ai fini di evitare cadute durante la marcia del mezzo. Grazie.
/si 'prεgano i dʒen'tili passed'dʒεri di affer'rare salda'mente un ele'mento del'l‿au̯tobus ai̯ 'fini di evi'tare ka'dute du'rante la 'martʃa del 'mεddzo 'grattsje/
si preg-ano i gentil-i passegger-i di afferr-are salda-mente un elemento de-llo autobus a-i fin-i di evit-are cadut-e durante la marcia de-l mezzo grazie
IMPERS pray-3PL DEF.MP gentle-MP passenger-PL of grab-INF firm-ADV INDEF.M element of-DEF.MS bus to-MP purpose-PL of avoid-INF fall-PL during DEF.FS journey of-DEF.MS vehicle thanks
Note that this form uses an impersonal particle, specifically a medio-passive reflexive, as if they didn't want to say who is requiring this.
How I would say this to a friend:
Tien(i)ti stretto!
(dropping the <i> is common around here, but I think it's non-standard everywhere else in Italy)
/'tjεniti 'stretto/
tien-i-ti strett-o
hold-IMPER-REFL.2SG firm-MS
How I would say this to someone I don't know:
Signore/a, Le consiglio di tenersi stretto/a.
/siɲ'ɲore le kon'siʎʎo di te'nersi 'stretto/
signore/a le consigli-o di tener-si strett-o/a
sir/madam 3SG.DAT.FORMAL advise-1SG to hold-REFL.3SG firm-MS/FS