We have a family plan for our phones, so our AP has an iPhone 5. If nothing else, I want her to be able to contact me by phone or text at any time. It's also critical for our family calendar as we all put appointments on there, GPS, camera for taking pictures of the kids. I have set up a linked family account on iTunes so she can download whatever apps I've already downloaded/paid for. We could not function as efficiently if she didn't have the smartphone, so I see it as a tool of the job, not primarily a benefit to her.
I have set up a generic au pair email address --
au.pair.ourlastname@gmail.com -- so that the recurring calendar events I've scheduled won't disappear when she leaves. I also use the share contact feature to set up all the contacts (schools, doctors, camps, friends, etc.). So the phone transfers to the new au pair and I don't have to set up everything all over again (been there, done that).
She can add her own personal email as an additional account that can be deleted when she leaves. She can also use WhatsApp or Snapchat to talk to her friends. She knows not to use data-heavy apps out and about unless she's on WiFi, but at home she or on WiFi she can use Netflix etc. She doesn't use it to call home -- instead she Skypes via WiFi.
I think divided out among all our family members who belong to the plan it's about $60 a month for her phone, which I think is reasonable. We all start getting text warnings from Verizon if we are close to or over our data limit. It's not my AP's phone that runs it over but she is cautious about data anyway.
I think the no romantic sleepover rule is a good way to phrase it. I also have a no romantic partner hanging out with the kids during work hours and a no romantic partner over at all unless we've met them first and approved. Our first AP had a lot of boyfriend drama (she had a lot of drama, period) and it was....unpleasant.