Foreword: I am focusing on DotA 2 since i stopped playing DotA 1 for quite a while now.
Offensive
Stuns will hold Morphling in place but wont interfere with his morph ability in any way.
Even Void's Chronosphere cannot stop his health gain.
Therefore you need a silence whether its an ability or an item to counter Morphling.
Important sidenote: Mantastyle does remove Orchid silence.
Naix with Orchid can be good because even if Morphling morphed into strength Naixwill deal more damage.
Clinkz with Orchid works well too because of his burst damage.
Silencer is very good because his Ultimate is not a target spell and global giving your team the opportunity to counter Morphling anytime.
(thanks to Decency for mentioning those 2 in his answer):
Ancient Apparation is another great pick vs Morphling because it prevents all kinds of healing including those from skills or items.
Lion's Mana Drain and Hex can destroy an illusion immediately. Combined with his long disable overall and his nuke he can be a threat at all times especially with Blink Dagger.
Defensive
To protect yourself against the enormous nuke of an Ethereal Blade-Morphling you should buy magic armor and a Mekansm. (Chen's Hand of God can be a substitute for Mekansm. But since you need to heal yourself quickly i would not rely on another players reaction. Having both is even better)
Furthermore BKB and spells like Repel help a lot.
Since EB-Morphling needs as much agility as possible to nuke hard he wont have much HP while nuking. This opens a window for a Blademail that can hit Morphling hard and enables the possibility for a good trade (your support for their midgame-carry)
Since Linkens Sphere is annoying you should pick heroes who can afford wasting a spell to disable his Linkens and have another good stun as a follow up.
For example: Bane, Lion, Shadow Shaman, Shadow Demon, Ogre, Rubik, Disruptor.