1. Ŝi vidis lin kuranta.
2. Ŝi vidis lin kurantan.
3. Ŝi vidis lin kuri.

Bear in mind that the kurantan in 2 is not an object predicate, but is used attributively.

The kuranta in 1 no doubt is an object predicate.

With 3 I am not quite sure, but I doubt that kuri alone is an object predicate.

I personally would never use 3 because of this:
If "Ŝi vidis lin kuri." meant "Ŝi vidis, ke li kuras/is.", then one might argue (or be tempted to believe) that "Ŝi volis lin helpi" meant "Ŝi volis, ke li helpu." and not "Ŝi volis doni helpon al li."

ECM verbs are mean buggers.

nornen:
Bear in mind that the kurantan in 2 is not an object predicate, but is used attributively.

The kuranta in 1 no doubt is an object predicate.

Yes, I see. Indeed, I don't think 2 & 3 are object predicates, either. I added them here because I'd like to know whether they were able to use or not.
I am still not sure if I can say like 3. But thank you very much for your answer!

Ŝi kuranta vidis lin

robbkvasnak:Ŝi kuranta vidis lin

Only this one is possible...?

"Ŝi vidis lin kuri."

I would advise not to use the infinitive here to say "She saw him run". The problem, as nornen indicates above, is that in infinitives these constructions often are more naturally parsed as pertaining to the main verb, not the object.

"Ŝi vidis lin kurantan."

This is the most natural and obvious way to put it, to my ear. "Kurantan" is attributive, not predicative, hence the meaning is: "She saw him, i.e. the running person."

"Ŝi vidis lin kuranta."

With the object predicate, the nuance is different now: This means "She saw/found him in the state of running." In other words, the focus is on her perception of his running, not on his running as being a descriptor of him. Analogous case: Mi trovis la libron enua, meaning that *I* found/perceived the book to be boring, rather than the book actually (necessarily) being boring, which would require enuan.

One common way to tackle structural analysis of phrases, is to transform them and to look at the result. Those transformations may include topicalization, questions or passivization.

What happens when we passivize the sentence Mi vidis lin kuri.
Is it ?Li estis vidita kuri?
Is it even possible to passivize this sentence?

@lunaris_filia: I saw in your profile that you have studied Latin. This transformation should be the equivalent of converting an accusativus cum infinitivo with an active verb (Romanos in Gallia vicisse dicit aliquis) to an nominativus cum infinitivo with a passive verb (Romani in Gallia vicisse dicuntur).

robbkvasnak:Ŝi kuranta vidis lin

I would parse this as "She, the one who was running, saw him." (Though kurante is better.) But not the meaning the original poster intended, I don't think.

nornen:One common way to tackle structural analysis of phrases, is to transform them and to look at the result. Those transformations may include topicalization, questions or passivization.

What happens when we passivize the sentence Mi vidis lin kuri.
Is it ?Li estis vidita kuri?
Is it even possible to passivize this sentence?

@lunaris_filia: I saw in your profile that you have studied Latin. This transformation should be the equivalent of converting an accusativus cum infinitivo with an active verb (Romanos in Gallia vicisse dicit aliquis) to an nominativus cum infinitivo with a passive verb (Romani in Gallia vicisse dicuntur).

Yes, that's what I am thinking about. I think in Latin we can say "(Ea) Eum vidit currere", which is "AcI". But when it is in Esperanto, I'd doubt whether it is possible or not. Thanks.

Tempodivalse:

"Ŝi vidis lin kuri."

I would advise not to use the infinitive here to say "She saw him run". The problem, as nornen indicates above, is that in infinitives these constructions often are more naturally parsed as pertaining to the main verb, not the object.

"Ŝi vidis lin kurantan."

This is the most natural and obvious way to put it, to my ear. "Kurantan" is attributive, not predicative, hence the meaning is: "She saw him, i.e. the running person."

"Ŝi vidis lin kuranta."

With the object predicate, the nuance is different now: This means "She saw/found him in the state of running." In other words, the focus is on her perception of his running, not on his running as being a descriptor of him. Analogous case: Mi trovis la libron enua, meaning that *I* found/perceived the book to be boring, rather than the book actually (necessarily) being boring, which would require enuan.

Ok. Thanks for your explanation!

Cf. PMEG:

Observaj k.s. ĉefverboj«

Kiam la ĉefverbo estas vidi, aŭdi, senti, imagi aŭ simile, povas aperi I-verbo, kiu estas perverba priskribo de la objekto de la ĉefverbo. Tiam tiu objekto estas senca subjekto de la I-verbo:

Ho, Marion, ke mi denove aŭdas vin paroli. BV.80 Paroli estas perverba priskribo de vin, la objekto de aŭdas. Senca subjekto de paroli estas vi.

Unfortunately I cannot share Bertilow's analysis of the infinitive being an object predicate. IMHO the direct object (i.e. complement) of aŭdas is vin paroli.

Bertilow: ? Ho, Marion, ke mi denove aŭdas [object: vin] [object predicate: paroli].
Nornen: ? Ho, Marion, ke mi denove aŭdas [object: vin paroli].

Rationale:
Compare the questions relating to the object predicate [kuranta] in I and the complex object [lin kuri] in II:
I. Mi vidis [lin] [kuranta]. --> Kiun vi vidis?
II. Mi vidis [lin kuri]. --> Kion vi vidis?

Compare the pronouns for the antecedents [lin] in I and [lin ŝteli kukon] in II:
I. Mi vidis [lin] [ŝtelanta kukon]. --> Mi ankaux vidis lin (=la ŝtelanton).
II. Mi vidis [lin ŝteli kukon]]. --> Mi ankaux vidis ĝin (=lian ŝteladon).

In summary I think there is indeed a difference between the use of a object predicate and a complex (verbal) object.
Mi vidis lin kuranta. = Mi vidis lin kaj li kuris.
Mi vidis lin kuri. = Mi vidis, ke li kuris.

An example where the difference may be stronger:
Mi vidis lin eliranta la oficejon. = Mi vidis lin, kaj tiu-momente li proksimigxis al la pordo. Tamen mi ne certas cxu li finfine eliris la oficejon.
Mi vidis lin eliri la oficejon. = Mi vidis, ke li eliris la oficejon. Sendube li eliris gxin.