I bring them back a [1] level lower than the lowest level PC in the party at the time. That way, they always stay relevant/comparable to the other PCs, but still suffer a "setback" for dying (particularly at higher levels)...to agree/piggyback on Umbran's point, there is some inducement NOT to die. Without having to go all the way back to "0" [level 1].
That said, how to proceed from character death is totally up to the players. If they want to roll up someone new, then fine: 1 level lower than the lowest current party member. If they decide they want that character back or their PC has enough potential in the story [and/or the party is wealthy enough and/or capable themselves, at higher levels] or there's some "grand scheme destiny" thing at work or whatever reason the players [as a whole, not just the PC's player, though they have final say] want to bring the character back to life. That's fine too.
I've had characters -at all levels- promise/make contracts/deals/etc... to bring each other back to life. Thus creating automatic side quest/story for the raising/collecting of funds (or promises/pacts of service), finding a cleric of appropriate power [or item or a fountain of life or whatever] to do it. Give the player of the deceased a [1 level lower than the lowest party member] NPC or roll up some temp character themselves to play in the interim [who can just be npc'd again or "leaves" for whatever reason or is killed off -or the player then plays 2 characters, if they want/can handle it- when the original pc is alive again].
Bringing them back as is, at their former level/the party's current level, leaves no room/interest/investment in NOT dying. Especially if you can roll up a new character who is higher than you were playing before?! That's REWARDING "getting dead." That is, to my D&D senses, completely the reverse of the intentions of the game.
Maybe these times "where tactics are things other people use" that ends with dead PCs should get the players to say, "Hmm. Maybe tactics are things WE should use. Maybe we won't end up dead if we did." Yeaaah, ya THINK?!
I would also raise the question/something to think about: if that's how you did things before/"back in the day", what's stopping you from doing it that way now? 5e certainly hasn't/doesn't (from what I've seen) tell you not to. I, personally, think making them start back at 1st gets problematic, as others have noted, beyond a 3rd or 4th level party. But, as others have noted, it's your table. Do whatever you think is best for your table.
I once had a party of about 3rd level PCs lose their "leader guy/fighter." The player really liked the PC and the background/story he'd come up with, family ties, etc. etc... The other PCs really liked this guy, had built various levels of relationships and loyalty with him, etc... They decided to bring him back, but [at only 3rd level] didn't have the money...and were put on a clock for how long the temple of the death goddess [which always required traveling to] could maintain the body/put the spirit's journey "on hold" to recall.
The player of the dead PC took on an NPC (who became another group favorite) and off they went. A couple good adventures later, the party was all around 5th, as I recall, before they had the necessary funds. The fighter was brought back at the 3rd level at which he had died. They carried on and he caught up fairly quickly, always staying within the level range -and surpassing some- of the other party members.
I've had mid-level characters do the same. High level parties where keeping Raise Dead in the daily spell-pool was practically like having potions of healing.
I would simply impose SOME kind of penalty for a "do over/next/restart" character and would definitely NOT let them re-enter the game at a level higher than they were. Otherwise, there's really no threat in the game. There is no impetus to not get killed. In effect, even though they're dying, they're all "Mary Sue'd" in...even if they die, they have no limitations. You're never really "gaining" anything as you play or [more importantly to me] receiving any consequences for your decisions. You're always just the same [or even better!]...regardless of the character you're playing. I would hate that.